8.2 Finding Complex Solutions of Polynomial Equations

Size: px
Start display at page:

Download "8.2 Finding Complex Solutions of Polynomial Equations"

Transcription

1 Locker LESSON 8. Finding Complex Solutions of Polynomial Equations Texas Math Standards The student is expected to: A.7.D Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods. Also A.7.B, A.7.E Mathematical Processes A.1.F The student is expected to analyze mathematical relationships to connect and communicate mathematical ideas. Language Objective.D.1,.I.,.E,.H.,.G Complete a Solving Polynomial Equations chart with a partner. Name Class Date 8. Finding Complex Solutions of Polynomial Equations Essential Question: What do the Fundamental Theorem of Algebra and its corollary tell you about the roots of the polynomial equation p(x) = 0 where p(x) has degree n? A.7.D Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods. Also A.7.B, A.7.E Explore Investigating the Number of Complex Zeros of a Polynomial Function You have used various algebraic and graphical methods to find the roots of a polynomial equation p (x) = 0 or the zeros of a polynomial function p (x). Because a polynomial can have a factor that repeats, a zero or a root can occur multiple times. The polynomial p (x) = x + 8 x + 1x + 18 = (x + ) (x + ) has - as a zero once and - as a zero twice, or with multiplicity. The multiplicity of a zero of p (x) or a root of p (x) = 0 is the number of times that the related factor occurs in the factorization. In this Explore, you will use algebraic methods to investigate the relationship between the degree of a polynomial function and the number of zeros that it has. Find all zeros of p (x) = x + 7x. Include any multiplicities greater than 1. p (x) = x + 7 x Factor out the GCF. p (x) = (x + 7) What are all the zeros of p(x)? 0 (mult. ), -7 x Resource Locker ENGAGE Essential Question: What do the Fundamental Theorem of Algebra and its corollary tell you about the roots of the polynomial equation p (x) = 0 where p (x) has degree n? The equation has exactly n complex roots provided that you count the multiplicities of the roots. Find all zeros of p (x) = x - 6. Include any multiplicities greater than 1. p (x) = x - 6 Factor the difference of two cubes. p (x) = (x - ) (x + x + 16 ) What are the real zeros of p (x)? Solve x + x + 16 = 0 using the quadratic formula. -b ± b - ac a ± i 1 ± - ± PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and what variables you might use to describe the amount of violence in a movie. Then preview the Lesson Performance Task. - ± i What are the non-real zeros of p (x)? - + i, - -i Module 8 7 Lesson Name Class Date 8. Finding Complex Solutions of Polynomial Equations Essential Question: What do the Fundamental Theorem of Algebra and its corollary tell you about the roots of the polynomial equation p(x) = 0 where p(x) has degree n? A.7.D Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods. Also A.7.B, A.7.E Explore Investigating the Number of Complex Zeros of a Polynomial Function You have used various algebraic and graphical methods to find the roots of a polynomial equation p (x) = 0 or the zeros of a polynomial function p (x). Because a polynomial can have a factor that repeats, a zero or a root can occur multiple times. The polynomial p (x) = x + 8 x + 1x + 18 = (x + ) (x + ) has - as a zero once and - as a zero twice, or with multiplicity. The multiplicity of a zero of p (x) or a root of p (x) = 0 is the number of times that the related factor occurs in the factorization. In this Explore, you will use algebraic methods to investigate the relationship between the degree of a polynomial function and the number of zeros that it has. Find all zeros of p (x) = x + 7x. Include any multiplicities greater than 1. p (x) = x + 7 x x 0 (mult. ), -7 Factor out the GCF. p (x) = (x + 7) What are all the zeros of p(x)? p (x) = x - 6 Find all zeros of p (x) = x - 6. Include any multiplicities greater than 1. Factor the difference of two cubes. p (x) = (x ) (x + + ) What are the real zeros of p (x)? Solve x + x + 16 = 0 using the quadratic formula. -b ± b - ac a ± ± - ± i What are the non-real zeros of p (x)? - x 16 - ± _ i i, - -i Resource Module 8 7 Lesson HARDCOVER PAGES 01 1 Turn to these pages to find this lesson in the hardcover student edition. 7 Lesson 8.

2 C Find all zeros of p (x) = x + x - x - 1x. Include any multiplicities greater than 1. p (x) = x + x - x - 1x Factor out the GCF. p (x) = x ( x ) + x - x - 1 Group terms to begin p (x) = x ( ( x + x ) - ( x + 1 )) factoring by grouping. Factor out common monomials. p (x) = x ( x (x + ) - (x + ) ) Factor out the common binomial. p (x) = x (x + ) ( x - ) Factor the difference of squares. p (x) = x (x + ) ( x + )( x - ) What are all the zeros of p (x)? 0, -, -, D Find all zeros of p (x) = x Include any multiplicities greater than 1. p (x) = x - 16 Factor the difference of squares. p (x) = ( x ) - ( x + ) Factor the difference of squares. p (x) = ( x + )( x - )( x + ) What are the real zeros of p (x)? -, EXPLORE Investigating the Number of Complex Zeros of a Polynomial Function INTEGRATE TECHNOLOGY Students have the option of completing the Explore activity either in the book or online. QUESTIONING STRATEGIES When would you need to use the quadratic formula to find a zero? When one of the factors of the polynomial is a non-factorable quadratic polynomial. Solve x + = 0 by taking square roots. x + = 0 x = - ± _ - ± i What are the non-real zeros of p (x)? -i, i Module 8 8 Lesson PROFESSIONAL DEVELOPMENT Learning Progressions Students have learned factoring techniques in earlier lessons, and a more general technique for finding zeros of polynomial functions and solutions of polynomial equations based on the Rational Zero/Root Theorem in the previous lesson. They have also learned how to use the quadratic formula to solve quadratic equations. In this lesson, students pull all these techniques together in order to understand and use the Fundamental Theorem of Algebra. Finding Complex Solutions of Polynomial Equations 8

3 INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Encourage students to look for patterns in their results. They can make connections between the degree of each polynomial and the number of zeros, and between a function s characteristics and their effects on the nature of its zeros. Students can also be prompted to make conjectures about the number of each type of zero (real and non-real) that could exist for polynomials of varying degrees. E Find all zeros of p (x) = x + 5 x + 6 x -x -8. Include multiplicities greater than 1. By the Rational Zero Theorem, possible rational zeros are ±1, ±, ±, and ±8. Use a synthetic division table to test possible zeros. m_ n The remainder is 0, so 1 is/is not a zero. p (x) factors as (x - 1) ( ). Test for zeros in the cubic polynomial. m_ n x + 6 x + 1x a zero. p (x) factors as (x - 1) (x + ) ( x ) + x +. The quadratic is a perfect square trinomial. So, p (x) factors completely as p (x) = (x - 1) (x + ). 1, - (mult. ) What are all the zeros of p (x)? F Complete the table to summarize your results from Steps A E. Polynomial Function in Standard Form p (x) = x + 7 x p (x) = x - 6 p (x) = x + x - x - 1x p (x) = x - 16 Polynomial Function Factored over the Integers p (x) = x (x + 7) p (x) = (x - ) ( x + x + 16) p (x) = x (x + ) (x + ) (x - ) p (x) = (x - ) (x + ) ( x + ) Real Zeros and Their Multiplicities 0 (mult. ) ; -7 0; -; -; -; Non-real Zeros and Their Multiplicities None - + i ; - - i None -i; i p (x) = x + 5 x + 6 x - x - 8 p (x) = (x - 1) (x + ) 1, - (mult. ) None Module 8 9 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activity Provide pairs of students with a fourth degree polynomial equation and a fifth degree polynomial equation. Have them work together to determine the number of possible combinations of types of roots for each equation. Then have them graph their equations, and use the graphs to help predict which combination of roots will be the correct combination for each function. Challenge them to solve the equations to verify their predictions. 9 Lesson 8.

4 Reflect 1. Examine the table. For each function, count the number of unique zeros, both real and non-real. How does the number of unique zeros compare with the degree? The number of unique zeros is less than or equal to the degree.. Examine the table again. This time, count the total number of zeros for each function, where a zero of multiplicity m is counted as m zeros. How does the total number of zeros compare with the degree? The total number of zeros is the same as the degree of the function.. Discussion Describe the apparent relationship between the degree of a polynomial function and the number of zeros it has. The number of zeros of a polynomial function is the same as the degree of the function when you include complex zeros and count the multiplicities of the zeros in the total. Explain 1 Applying the Fundamental Theorem of Algebra to Solving Polynomial Equations The Fundamental Theorem of Algebra and its corollary summarize what you have observed earlier while finding rational zeros of polynomial functions and in completing the Explore. The Fundamental Theorem of Algebra Every polynomial function of degree n 1 has at least one zero, where a zero may be a complex number. EXPLAIN 1 Applying the Fundamental Theorem of Algebra to Solving Polynomial Equations INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Substantiate The Fundamental Theorem of Algebra and its corollary by applying them to solutions of linear equations and easily factorable quadratic equations, with which students are familiar. Include examples of quadratic equations that have roots with multiplicity of. Corollary: Every polynomial function of degree n 1 has exactly n zeros, including multiplicities. Because the zeros of a polynomial function p (x) give the roots of the equation p (x) = 0, the theorem and its corollary also extend to finding all roots of a polynomial equation. Example 1 Solve the polynomial equation by finding all roots. x - 1 x - x + 0 = 0 The polynomial has degree, so the equation has exactly roots. x - 1 x - x + 0 = 0 Divide both sides by. x - 6 x - 17x + 10 = 0 Group terms. ( x - 6 x ) - (17x - 10) = 0 Factor out common monomials. x (x - 6) - 17 (x - 6) = 0 Factor out the common binomial. ( x - 17) (x - 6) = 0 One root is 6. Solving x - 17 = 0 gives x = 17, or ± _ 17. The roots are - _ 17, _ 17, and 6. Module 8 0 Lesson DIFFERENTIATE INSTRUCTION Communicating Math Understanding the concept of the degree of a polynomial is important in applying the Fundamental Theorem of Algebra and its corollary. Students (especially English language learners) may benefit from a rigorous review of finding degrees of polynomials written in standard form, factored form, and with terms in varying orders of degree. Focus on polynomials that contain only single-variable monomials. Check that students can explain how to find the degree of the polynomial for each of the different forms. Finding Complex Solutions of Polynomial Equations 0

5 QUESTIONING STRATEGIES If, after using synthetic substitution to test all possible rational roots of a cubic equation, you find only one root of the equation, can you conclude that the remaining roots are imaginary? Explain. No. The remaining roots may be imaginary or they may be irrational. B x - 6 x - 7 = 0 The polynomial has degree, so the equation has exactly roots. Notice that x - 6 x - 7 has the form u - 6u - 7, where u = x. So, you can factor it like a quadratic trinomial. x - 6 x - 7 = 0 Factor the trinomial. ( x - ) ( x + ) = 0 Factor the difference of squares. ( x + ) (x - ) ( x + ) = 0 The real roots are - and. Solving x + = 0 gives x = -, or 9 ± _ - = ± i. The roots are -,, -i, i. Reflect. Restate the Fundamental Theorem of Algebra and its corollary in terms of the roots of equations. Theorem: For every polynomial of degree n 1, the equation p (x) = 0 has at least one root, where a root may be a complex number. Corollary: For every polynomial of degree n 1, the equation p (x) = 0 has exactly n roots, when you include multiplicity. Your Turn Solve the polynomial equation by finding all roots x - 7 = 0 6. p (x) = x - 1 x + 55 x - 91x (x - ) ( x + 6x + 9) = 0 x - = 0 _ x + 6x + 9 = 0 -(6) ± (6) - ()(9) () -6 ± -108 = -6 ± 6i ± i, or - _ ± _ i The roots are _ -, + i, and - - i. x ( x - 1 x + 55x - 91) = 0 One root is 0. Possible rational roots: ±1, ±7, ±1, ±91. Use synthetic division to test for roots. A second root is 7. Solve x - 6x + 1 = 0. -(-6) ± (-6) - (1)(1) 1 6 ± -16 = 6 ± i ± i The roots are 0, 7, + i, and - i. Module 8 1 Lesson LANGUAGE SUPPORT Communicate Math Have students work in pairs. Have them write the theorems in this module for solving polynomial equations, the Rational Zero Theorem, Rational Roots Theorem, and the Fundamental Theorem of Algebra, and then work together to explain the theorems in their own words. Then have students write the explanations and give an example for each theorem. 1 Lesson 8.

6 Explain Writing a Polynomial Function From Its Zeros You may have noticed in finding roots of quadratic and polynomial equations that any irrational or complex roots come in pairs. These pairs reflect the ± in the quadratic formula. For example, for any of the following number pairs, you will never have a polynomial equation for which only one number in the pair is a root. _ 5 and - _ 5 ; and 1-7 ; i and -i; + 1i and - 1i; _ _ 6 i and 11_ 6-1_ 6 i The irrational root pairs a + b c and a - b c are called irrational conjugates. The complex root pairs a + bi and a - bi are called complex conjugates. Irrational Root Theorem If a polynomial p (x) has rational coefficients and a + b c is a root of the equation p (x) = 0, where a and b are rational and c is irrational, then a - b c is also a root of p (x) = 0. Complex Conjugate Root Theorem If a + bi is an imaginary root of a polynomial equation with real-number coefficients, then a - bi is also a root. Because the roots of the equation p (x) = 0 give the zeros of a polynomial function, corresponding theorems apply to the zeros of a polynomial function. You can use this fact to write a polynomial function from its zeros. Because irrational and complex conjugate pairs are a sum and difference of terms, the product of irrational conjugates is always a rational number and the product of complex conjugates is always a real number. ( - _ 10 ) ( + _ 10 ) = - ( _ 10 ) = - 10 = -6 (1 - i _ ) (1 + i _ ) = 1 - (i _ ) = 1 - (-1) () = EXPLAIN Writing a Polynomial Function From its Zeros QUESTIONING STRATEGIES If one zero of a fourth degree polynomial function is rational, what must be true about the other three zeros? One of the three must also be rational. The other two could be either irrational conjugates or imaginary conjugates. Is it possible for a fifth degree polynomial equation to have no real zeros? Explain. No. Since imaginary zeros occur in conjugate pairs, there could be at most imaginary zeros. Therefore, at least one zero must be real. Example Write the polynomial function with least degree and a leading coefficient of 1 that has the given zeros. 5 and + _ 7 Because irrational zeros come in conjugate pairs, - _ 7 must also be a zero of the function. Use the zeros to write a function in factored form, then multiply to write it in standard form. Multiply the first two factors using FOIL. p (x) = x - ( + _ 7 ) x - ( - _ 7 ) (x - 5) Multipy the conjugates. = x - ( - _ 7 ) x - ( + _ 7 ) x + (9-7) (x - 5) Combine like terms. = x + (- + _ _ 7 ) x + (-19) (x - 5) Simplify. = x - 6x - 19 (x - 5) = x - ( - _ 7 ) x - ( + _ 7 ) x + ( + _ 7 )( - _ 7 ) (x - 5) Distributive property = x ( x - 6x - 19) - 5 ( x - 6x - 19) AVOID COMMON ERRORS Students may make errors when multiplying factors of the form (x - a), where a is an irrational number such as + or an imaginary number such as 1 - i. Encourage them to multiply each of these types of factors with the factor that contains the conjugate of the irrational or imaginary number first, and show them how to use grouping to make the multiplication easier. Multiply. = x - 6 x - 19x - 5 x + 0x + 95 Combine like terms. = x - 11 x + 11x + 95 The polynomial function is p (x) = x - 11 x + 11x Module 8 Lesson Finding Complex Solutions of Polynomial Equations

7 INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Have students discuss how they could write the rule for a third degree polynomial function whose graph passes through (1 +, 0) and the origin. Then have them find the function, and use a graphing calculator to check their work. B, and 1- i Because complex zeros come in conjugate pairs, must also be a zero of the function. Use the zeros to write a function in factored form, then multiply to write it in standard form. p (x) = x - (1 + i) x - ( 1 - i ) (x - ) (x - ) Multiply the first = x - (1 - i) x - two factors using FOIL. ( 1 + i ) x + (1 + i) (1 - i) (x - ) (x - ) Multipy the conjugates. Combine like terms. = x + (-1 + i i) x + (x - ) (x - ) Simplify. = ( ) (x - ) (x - ) Multipy the binomials. = ( x - x + ) Distributive property = x ( x - 5x + 6) - x ( x - 5x + 6) + ( x - 5x + 6) Multipy. = ( x - 5 x + 6 x ) + (- x + 10 x - 1x) + ( x - 10x + 1) Combine like terms. = = 1 + i x - (1 - i) x - (1 + i) x + ( 1 - ( x - x + ( x - 5x + 6) x - 7 x + 18 x - x The polynomial function is p (x) = x - 7 x + 18 x - x + 1. )) (x - ) (x - ) Reflect 7. Restate the Irrational Root Theorem in terms of the zeros of polynomial functions. If a polynomial function p (x) has rational coefficients and a + b c is a zero of the function, where a and b are rational and c is irrational, then a - b c is also a zero of p (x). 8. Restate the Complex Conjugates Zero Theorem in terms of the roots of equations. If a + bi is an imaginary zero of a polynomial function p (x) with real-number coefficients, then a - bi is also a zero of p (x). Module 8 Lesson Lesson 8.

8 Your Turn Write the polynomial function with the least degree and a leading coefficient of 1 that has the given zeros i and - 7 _ The polynomial function must also have - i and + 7 as zeros. p (x) = x - ( + i) x - ( - i) x - ( + 7 ) x - ( - 7 ) = x - ( - i) x - ( + i) x + ( + i) ( - i) x - ( - 7 x + ( + 7 ) ( - 7 ) = x - ( - i) x - ( + i) x + ( - 9 (-1)) x - ( - 7 ) x - ( + 7 ) x - ( + 7 x + (16-9 ) = x + (- + i - - i) x + 1 x + ( ) x - 8 = ( x - x + 1) ( x - 8x - 8) = x ( x - 8x - 8) -x ( x - 8x - 8) + 1 ( x - 8x - 8) = ( x - 8 x - 8 x ) + (- x + x + 8x) + (1 x - 10x ) = x - 1 x - 7 x + x The polynomial function is p (x) = x - 1 x - 7 x + x ) ) EXPLAIN Solving a Real-World Problem by Graphing Polynomial Functions INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Lead students to recognize that the solution of the problem is not a zero of either p (x) or q (x) ; however, it is a zero of the difference function p (x) - q (x). This can be confirmed from the graphs of the three functions. Explain Solving a Real-World Problem by Graphing Polynomial Functions You can use graphing to help you locate or approximate any real zeros of a polynomial function. Though a graph will not help you find non-real zeros, it can indicate that the function has non-real zeros. For example, look at the graph of p (x) = x - x -. The graph intersects the x-axis twice, which shows that the function has two real zeros. By the corollary to the Fundamental Theorem of Algebra, however, a fourth degree polynomial has four zeros. So, the other two zeros of p (x) must be non-real. The zeros are - _, _, i, and -i. (A polynomial whose graph has a turning point on the x-axis has a real zero of even multiplicity at that point. If the graph bends at the x-axis, there is a real zero of odd multiplicity greater than 1 at that point.) - y x Module 8 Lesson Finding Complex Solutions of Polynomial Equations

9 QUESTIONING STRATEGIES Why do the methods shown in Parts A and B produce the same solution? When you solve the equation p (x) = q (x), you are finding the value of x for which the two functions are equal. Since p(x) is equal to q (x) at this value of x, this is the value that would make their difference, p (x) - q (x), equal to 0. The following polynomial models approximate the total oil consumption C (in millions of barrels per day) for North America (NA) and the Asia Pacific region (AP) over the period from 001 to 011, where t is in years and t = 0 represents 001. C NA (t) = t t + 0. t - 0.9t +.6 C AP (t) = t t + 1.t Use a graphing calculator to plot the functions and approximate the x-coordinate of the intersection in the region of interest. What does this represent in the context of this situation? Determine when oil consumption in the Asia Pacific region overtook oil consumption in North America using the requested method. Graph Y1 = x x + 0. x - 0.9x +.6 and Y = x x + 1.x Use the Calc menu to find the point of intersection. Here are the results for Xmin = 0, Xma 10, Ymin = 0, Yma 0. (The graph for the Asia Pacific is the one that rises upward on all segments.) The functions intersect at about 5, which represents the year 006. This means that the models show oil consumption in the Asia Pacific equaling and then overtaking oil consumption in North America about 006. Find a single polynomial model for the situation in Example A whose zero represents the time that oil consumption for the Asia Pacific region overtakes consumption for North America. Plot the function on a graphing calculator and use it to find the x-intercept. Let the function C D (t) represent the difference in oil consumption in the Asia Pacific and North America. the time that consumption is equal A difference of 0 indicates. C D (t) = C AP (t) - C NA (t) = t t + 1.t (0.009 t t + 0. t - 0.9t +.6) Remove parentheses and rearrange terms. = t t t t - 0. t + 1.t + 0.9t Combine like terms. Round to three significant digits. = t t t + 1.7t -.50 Graph C D (t) and find the x-intercept. (The graph with Ymin = -, Yma 6 is shown.) Within the rounding error, the results for the x-coordinate of the intersection of C NA (t) and C AP (t) and the x-intercept of C D (t) are the same. Module 8 5 Lesson 5 Lesson 8.

10 Your Turn 10. An engineering class is designing model rockets for a competition. The body of the rocket must be cylindrical with a cone-shaped top. The cylinder part must be 60 cm tall, and the height of the cone must be twice the radius. The volume of the payload region must be 558π cm in order to hold the cargo. Use a graphing calculator to graph the rocket s payload volume as a function of the radius x. On the same screen, graph the constant function for the desired payload. Find the intersection to find x. Let V represent the volume of the payload region. V = V cone + V cylinder 1 V (x) = π x (x) + π x (60) = Elaborate π x + 60π x To find _ x when the volume is 558π, graph y = π x + 60π x and y = 558π and find the points of intersection. Because the radius must be positive, the radius of the rocket is cm. 11. What does the degree of a polynomial function p(x) tell you about the zeros of the function or the roots of the equation p (x) = 0? The degree tells you how many zeros or roots there are when you include complex zeros or roots and count the multiplicities of repeated zeros or roots. 1. A polynomial equation of degree 5 has the roots 0.,, 8, and 10.6 (each of multiplicity 1). What can you conclude about the remaining root? Explain your reasoning. The remaining root must be rational. This is because any irrational roots or imaginary roots always occur in conjugate pairs. So, if there were an irrational or imaginary root, there would have to be two of them. 1. Discussion Describe two ways you can use graphing to determine when two polynomial functions that model a real-world situation have the same value. You can graph both functions on the same coordinate grid and find the x-value of any point where the two graphs intersect. Also, you can form a new function that is the difference of the two original functions. The x-intercepts of the graph of this function will also be the x-values where the original functions have the same value. 1. Essential Question Check-In What are possible ways to find all the roots of a polynomial equation? By the corollary to the Fundamental Theorem of Algebra, you know that the number of roots equals the degree of the equation. You can factor when possible, and use the Rational Root theorem along with the Zero Product Property to find rational roots. You can use the quadratic formula to find irrational or complex roots. Module 8 6 Lesson ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Lead students to the generalization that a polynomial function of odd degree must have an odd number (counting repeated zeros) of real zeros and, in particular, must have at least one real zero. QUESTIONING STRATEGIES A fourth degree polynomial function has only the zeros -,, and. How can this be true given the requirement of the Corollary of the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n zeros? One of the zeros must occur twice. The corollary requires that repeated zeros be counted multiple times. INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Ask students to discuss the possibility of two polynomial functions that model a real-world situation having more than one value for which they are equal. Have them discuss the implications of this situation on the graphs of the functions and on the graph of the difference function. SUMMARIZE THE LESSON How can you use the Fundamental Theorem of Algebra, its corollary, and the Irrational Conjugates and Complex Conjugates Theorems to determine the possible combinations of types of zeros of a polynomial function? You can use the Fundamental Theorem of Algebra and its corollary to find the total number of zeros of the function. Then you can use the fact that irrational and imaginary zeros occur in conjugate pairs to determine the possible combinations. Finding Complex Solutions of Polynomial Equations 6

11 EVALUATE Evaluate: Homework and Practice Find all zeros of p (x). Include any multiplicities greater than 1. Online Homework Hints and Help Extra Practice ASSIGNMENT GUIDE Concepts and Skills Explore Investigating the Number of Complex Zeros of a Polynomial Function Example 1 Applying the Fundamental Theorem of Algebra to Solving Polynomial Equations Example Writing a Polynomial Function From its Zeros Example Solving a Real-World Problem by Graphing Polynomial Functions QUESTIONING STRATEGIES Practice Exercises 1 Exercises Exercises 5 8 Exercises 9 11 How does the Rational Zero Theorem help you find zeros that are not rational? The Rational Zero Theorem can be used to identify rational zeros and the corresponding factors. Then, other methods, such as the quadratic formula, may be used to find other zeros that are irrational or imaginary. 1. p (x) = x - 10 x + 10x -. p (x) = x - x + x - 1 Possible rational zeros are ±1, ±, ±, ± 1_, ± _, ± _. is a zero. p (x) = (x - ) ( x - x + ) Solve x - x + = 0. - (-) ± (-) - ()() () ± -8 ± i ± i = = 6 6 The zeros of p (x) are, + i, and - i. Solve the polynomial equation by finding all roots. p (x) = x - x + x - 1 = ( x - x ) + (x - 1) = x (x - ) + (x - ) = ( x + ) (x - ) is a zero. Solve x + = 0. x = - ± - = ±i The zeros of p (x) are, -i, and i.. x - x + 8x - 1 = 0. x - 5 x + x + 0 ( x - x ) + (8x - 1) = 0 x (x - ) + (x - ) = 0 (x + ) (x - ) = 0 _ x + = 0 x - = 0 - ± - = ±i The roots are _, -i, and i. x (x - 5 x + x + 1) = 0 0 is a root. Possible rational roots are 1 and is a root. x (x - 1) ( x - x - 1) = 0 Solve x - x - 1 = 0. - (-) ± (-) - (1)(-1) (1) ± 0 = ± 5 = ± 5 The roots are 0, 1, + 5, and - 5. Module 8 7 Lesson Exercise Depth of Knowledge (D.O.K.) Mathematical Processes 1 8 Skills/Concepts 1.F Analyze relationships 9 10 Skills/Concepts 1.D Multiple representations 11 Skills/Concepts 1.A Everyday life 1 Strategic Thinking 1.F Analyze relationships 1 1 Strategic Thinking 1.F Analyze relationships 15 Strategic Thinking 1.F Analyze relationships 7 Lesson 8.

12 Write the polynomial function with least degree and a leading coefficient of 1 that has the given zeros. 5. 0, _ 5, and 6. i,, and - Because irrational zeros come in conjugate pairs, - 5 must also be a zero. p (x) = x (x - 5 ) (x + 5 )(x - ) = x ( x - 5) (x - ) = x ( x - x - 5x + 10) = x - x - 5 x + 10x 7. 1, -1 (multiplicity ), and i 8. (multiplicity of ) and i Because complex zeros come in conjugate pairs, -i must also be a zero. p (x) = (x - ) (x + ) (x - i) (x + i) = ( x - ) ( x + 16) = x + 1 x - 6 Because complex zeros come in conjugate pairs, -i must also be a zero. p (x) = (x - 1) (x + 1) (x - i) (x + i) = (x - 1) (x + 1) (x + 1) (x - i) (x + i) = ( x - 1) ( x + x + 1) ( x + 9) = ( x + 8 x - 9) ( x + x + 1) = x ( x + x + 1) + 8 x ( x + x + 1) - 9 ( x + x +1) = x 6 + x 5 + x + 8 x + 16 x + 8 x - 9 x - 18x - 9 = x 6 + x x + 16 x - x - 18x - 9 Because complex zeros come in conjugate pairs, -i must also be a zero. p (x) = (x - ) (x - i) (x + i) = ( x - 6x + 9) ( x + 9) = x ( x - 6x + 9) + 9 ( x - 6x + 9) = x - 6 x + 9 x + 9 x - 5x + 81 = x - 6 x + 18 x - 5x + 81 AVOID COMMON ERRORS Students often make sign errors when writing factors for zeros or roots that are irrational, such as - 5, or imaginary, such as + i. Encourage them to use parentheses within parentheses when writing the factors, and to be careful to apply the distributive property when removing the parentheses or regrouping the terms. INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Have students discuss why irrational roots of a polynomial equation with rational coefficients must occur in conjugate pairs. Have them consider the resulting polynomial if, for example, only one of three factors of a cubic polynomial equation contained an irrational number. Module 8 8 Lesson Finding Complex Solutions of Polynomial Equations 8

13 VISUAL CUES Have students graph several of the functions using a graphing calculator to provide a visual connection between each type of zero (rational, irrational, and imaginary), and its representation on the graph of the function. Help students to see how irrational zeros can be approximated from x-intercepts. Lead them to observe that a function that has only imaginary zeros has no x-intercepts. CRITICAL THINKING Students may be interested to find that they can test irrational and imaginary zeros of a polynomial function using synthetic substitution. Encourage them to use this process to check their work. 9. Forestry Height and trunk volume measurements from 10 giant sequoias between the heights of 0 and 75 feet in California give the following model, where h is the height in feet and V is the volume in cubic feet. V (h) = 0.11 h h + 1,00h - 1,67,00 The President tree in the Giant Forest Grove in Sequoia National Park has a volume of about 5,100 cubic feet. Use a graphing calculator to plot the function V (h) and the constant function representing the volume of the President tree together. (Use a window of 0 to 75 for X and 0,000 to 55,000 for Y.) Find the x-coordinate of the intersection of the graphs. What does this represent in the context of this situation? 10. Business Two competing stores, store A and store B, opened the same year in the same neighborhood. The annual revenue R (in millions of dollars) for each store t years after opening can be approximated by the polynomial models shown. R A (t) = (- t + 1 t - 77 t + 600t + 1,650) R B (t) = (- t + 6 t t + 68t + 90) The x-coordinate of the intersection gives the model s predicted height for a tree with the volume of the President tree. This predicted height is about 65 feet. Image Credits: RichardBakerUSA/Alamy Using a graphing calculator, graph the models from t = 0 to t = 10, with a range of 0 to for R. Find the x-coordinate of the intersection of the graphs, and interpret the graphs. Graph Y1 = (- x + 1 x - 77 x + 600x + 1,650) for R A. Graph Y = (- x + 6 x x + 68x + 90) for R B. Then find the point of intersection. The functions intersect at 9, which corresponds to having the same annual revenue 9 years after the stores opened. Module 8 9 Lesson 9 Lesson 8.

14 11. Personal Finance A retirement account contains cash and stock in a company. The cash amount is added to each week by the same amount until week, then that same amount is withdrawn each week. The functions shown model the balance B (in thousands of dollars) over the course of the past year, with the time t in weeks. B C (t) = -0.1 t B S (t) = t t t t Use a graphing calculator to graph both models (Use 0 to 0 for range.). Find the x-coordinate of any points of intersection. Then interpret your results in the context of this situation. The graphs intersect at x-values of about 8 and 7. This means that at those weeks of the year, the cash balance and stock balance in the account were the same. LANGUAGE SUPPORT Connect Vocabulary Remind students that they learned complex numbers have a real and an imaginary part. The complex conjugate of a + bi is a - bi, and similarly the complex conjugate of a - bi is a + bi. This consists of changing the sign of the imaginary part of a complex number. The real part is left unchanged. 1. Match the roots with their equation. A. 1 A, B, E, F x + x + x + x - 8 = 0 B. - A, B, C, D x - 5 x + = 0 C. D. -1 E. i F. -i x + x + x + x - 8 = 0 in factored form is (x - 1) (x + ) ( x + ) = 0. Roots are 1, -, i, and -i. x - 5 x + = 0 in factored form is (x + 1) (x - 1) (x + ) (x - ) = 0. Roots are -1, 1, -, and. Module 8 0 Lesson Finding Complex Solutions of Polynomial Equations 0

15 PEER-TO-PEER DISCUSSION Ask students to discuss with a partner why, although the Rational Root Theorem can always be used to help find the roots of a cubic equation, it may not be useful for finding the roots of a fourth degree polynomial equation. Since a cubic equation has three roots, at least one of them will be rational (since irrational and imaginary roots occur in conjugate pairs). The other two roots, no matter what type, can be found by factoring or by using the quadratic formula. A fourth degree equation will have four roots, none of which may be rational, so the Rational Root Theorem may not be of help. JOURNAL Have students describe how they would go about finding the roots of a fifth degree polynomial equation if they know that at least two of the roots are rational. H.O.T. Focus on Higher Order Thinking 1. Draw Conclusions Find all of the roots of x 6 5 x 15x + 15,65 = 0. (Hint: Rearrange the terms with a sum of cubes followed by the two other terms.) ( x ,65) - 5 x - 65 x = 0 ( x ) x - 65 x = 0 ( x + 5) ( x - 5 x + 65) - 5 x ( x + 5) = 0 ( x + 5) ( x - 5 x x ) = 0 ( x + 5) ( x -50 x + 65) = 0 ( x + 5) ( x - 5) = 0 ( x + 5) (x+ 5) (x - 5) = 0 The roots are -5 and 5, each with multiplicity, and -5i and 5i. 1. Explain the Error A student is asked to write the polynomial function with least degree and a leading coefficient of 1 that has the zeros 1 + i, 1 - i, _, and -. The student writes the product of factors shown, and multiplies them together to obtain p (x) = x + (1 - _ ) x - ( + _ )x + (6 + _ ) x - 6 _. What error did the student make? What is the correct function? The function must have 5 zeros. The zero must be paired with its conjugate, -. p (x) = x - (1 + i) x - (1 - i) (x - ) (x + )(x + ) = x - (1 - i) x -(1 + i) x + (1 + i) (1 - i) ( x - ) (x + ) = x + (-1 + i -1 - i) x + (1 -(-1)) ( x + x -x - 6) = ( x - x + ) ( x + x - x - 6) = ( x 5 + x - x - 6 x ) + (- x - 6 x + x + 1x) + ( x + 6 x - x - 1) = x 5 + x - 6x + x + 8x Critical Thinking What is the least degree of a polynomial equation that has i as a root with a multiplicity of, and - _ as a root with multiplicity? Explain. The least degree is 10. Since i is a root times, then -i must also be a root times. Since - is a root times, then + must also be a root times, and = 10. Module 8 1 Lesson 1 Lesson 8.

16 Lesson Performance Task In 198 the MPAA introduced the PG-1 rating to their movie rating system. Recently, scientists measured the incidences of a specific type of violence depicted in movies. The researchers used specially trained coders to identify the specific type of violence in one half of the top grossing movies for each year since The trend in the average rate per hour of 5-minute segments of this type of violence in movies rated G/PG, PG-1, and R can be modeled as a function of time by the following equations: V G/PG (t) = t V PG-1 (t) = t t + 0.6t V R (t) =.15 V is the average rate per hour of 5-minute segments containing the specific type of violence in movies, and t is the number of years since a. Interestingly, in 1985 or t = 0, V G/PG (0) > V PG-1 (0). Can you think of any reasons why this would be true? b. What do the equations indicate about the relationship between V G/PG (t) and V PG-1 (t) as t increases? c. Graph the models for V G/PG (t) and V PG-1 (t) and find the year in which V PG-1 (t) will be greater than V G/PG (t). a. Possible answers include but are not limited to The rating of PG-1 was poorly understood by the people responsible for rating the films. Films released in the years immediately following 1985 had been scripted, filmed, and/or edited before the rating was fully understood by the film studios, so they hadn t separated the specific type of violence out of the G/PG movies. b. The equations indicate that as t increases, V PG-1 (t) will eventually be greater than V G/PG (t). V G/PG (t) is a linear function with a negative first term so its end behavior on the right is decreasing to negative infinity while the leading term of V PG-1 (t) is positive, so its end behavior on the right is increasing to infinity. c. The functions intersect at a value of t, which indicates that the average rate per hour of 5-minute segments of violence in movies rated PG-1 first surpassed the average hourly rate in movies rated G/PG in Rate per Hour 1 V(t) V R V G/PG Years Since 1985 V PG 1 t Connect Vocabulary Students may not be familiar with the abbreviations of the movie rating system. Explain that the abbreviations indicate how appropriate the movie is for difference audiences. A G rating means the movie is for General audiences. A PG rating means Parental Guidance is suggested. A PG-1 rating means Parental Guidance is suggested and the movie may not be appropriate for children under age 1. An R rating means entrance is Restricted; an adult must accompany children under 17. AVOID COMMON ERRORS Students may think that the models V (t) give the total amount of violence in a movie. Ask students what the units of V (t) are. number of 5-minute segments per hour Ask students how to calculate the total minutes of violence in a movie. Multiply V (t) by 5 and then multiply by the length of the movie in hours. INTEGRATE MATHEMATICAL PROCESSES Focus on Communication Discuss with students why V PG-1 increases to infinity as t increases. Ask them if it makes sense that V PG-1 becomes greater than V R and whether they think this will actually happen. Have students explain how they could create a model that would more accurately predict V PG-1 for future years.. Module 8 Lesson EXTENSION ACTIVITY Have students research the top-grossing movie for each year since 1985 and whether it was rated G, PG, PG-1, or R. Have students discuss whether the success of a movie is related to its rating. Ask them if they think the amount of violence in a movie makes it more or less popu lar. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Finding Complex Solutions of Polynomial Equations

7.2 Multiplying Polynomials

7.2 Multiplying Polynomials Locker LESSON 7. Multiplying Polynomials Teas Math Standards The student is epected to: A.7.B Add, subtract, and multiply polynomials. Mathematical Processes A.1.E Create and use representations to organize,

More information

6.2 Multiplying Polynomials

6.2 Multiplying Polynomials Locker LESSON 6. Multiplying Polynomials PAGE 7 BEGINS HERE Name Class Date 6. Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of epression is the result? Common

More information

Essential Question: What is a complex number, and how can you add, subtract, and multiply complex numbers? Explore Exploring Operations Involving

Essential Question: What is a complex number, and how can you add, subtract, and multiply complex numbers? Explore Exploring Operations Involving Locker LESSON 3. Complex Numbers Name Class Date 3. Complex Numbers Common Core Math Standards The student is expected to: N-CN. Use the relation i = 1 and the commutative, associative, and distributive

More information

12.2 Simplifying Radical Expressions

12.2 Simplifying Radical Expressions x n a a m 1 1 1 1 Locker LESSON 1. Simplifying Radical Expressions Texas Math Standards The student is expected to: A.7.G Rewrite radical expressions that contain variables to equivalent forms. Mathematical

More information

11.3 Solving Radical Equations

11.3 Solving Radical Equations Locker LESSON 11. Solving Radical Equations Common Core Math Standards The student is expected to: A-REI. Solve simple rational and radical equations in one variable, and give examples showing how extraneous

More information

!!! 1.! 4x 5 8x 4 32x 3 = 0. Algebra II 3-6. Fundamental Theorem of Algebra Attendance Problems. Identify all the real roots of each equation.

!!! 1.! 4x 5 8x 4 32x 3 = 0. Algebra II 3-6. Fundamental Theorem of Algebra Attendance Problems. Identify all the real roots of each equation. Page 1 of 15 Fundamental Theorem of Algebra Attendance Problems. Identify all the real roots of each equation. 1. 4x 5 8x 4 32x 3 = 0 2. x 3 x 2 + 9 = 9x 3. x 4 +16 = 17x 2 Page 2 of 15 4. 3x 3 + 75x =

More information

Finding Complex Solutions of Quadratic Equations

Finding Complex Solutions of Quadratic Equations COMMON CORE y - 0 y - - 0 - Locker LESSON 3.3 Finding Comple Solutions of Quadratic Equations Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the

More information

16.2 Solving Exponential Equations

16.2 Solving Exponential Equations Locker LESSON 16.2 Solving Exponential Equations Texas Math Standards The student is expected to: A2.5.D Solve exponential equations of the form y = ab x where a is a nonzero real number and b is greater

More information

11.1 Inverses of Simple Quadratic and Cubic Functions

11.1 Inverses of Simple Quadratic and Cubic Functions Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,

More information

Essential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically

Essential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically 6 7 6 y 7 8 0 y 7 8 0 Locker LESSON 1 1 Using Graphs and Properties to Solve Equations with Eponents Common Core Math Standards The student is epected to: A-CED1 Create equations and inequalities in one

More information

17.1 Understanding Polynomial Expressions

17.1 Understanding Polynomial Expressions COMMON CORE 4 a b Locker x LESSON Common Core Math Standards The student is expected to: COMMON CORE A-SSE.A.a Interpret parts of an expression, such as terms, factors, and coefficients. Also A-SSE.A.b,

More information

4.1 Understanding Polynomial Expressions

4.1 Understanding Polynomial Expressions 4 a b Locker x LESSON 4. Understanding Polynomial Expressions Common Core Math Standards The student is expected to: A-SSE.a Interpret parts of an expression, such as terms, factors, and coefficients.

More information

Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions COMMON CORE Locker LESSON 9.1 Adding and Subtracting Rational Epressions Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions?

More information

8.2 Graphing More Complicated Rational Functions

8.2 Graphing More Complicated Rational Functions 1 Locker LESSON 8. Graphing More Complicated Rational Functions PAGE 33 Name Class Date 8. Graphing More Complicated Rational Functions Essential Question: What features of the graph of a rational function

More information

15.2 Graphing Logarithmic

15.2 Graphing Logarithmic _ - - - - - - Locker LESSON 5. Graphing Logarithmic Functions Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes on the graphs of f () = b and f () = log b

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area

More information

Theorems About Roots of Polynomial Equations. Rational Root Theorem

Theorems About Roots of Polynomial Equations. Rational Root Theorem 8-6 Theorems About Roots of Polynomial Equations TEKS FOCUS TEKS (7)(E) Determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum

More information

Lesson #33 Solving Incomplete Quadratics

Lesson #33 Solving Incomplete Quadratics Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique

More information

6. 2 Multiplying Polynomials

6. 2 Multiplying Polynomials Name Class Date 6. 2 Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of expression is the result? Explore Analyzing a Visual Model for Polynomial Multiplication

More information

Theorems About Roots of Polynomial Equations. Theorem Rational Root Theorem

Theorems About Roots of Polynomial Equations. Theorem Rational Root Theorem - Theorems About Roots of Polynomial Equations Content Standards N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. Also N.CN.8 Objectives To solve equations using the

More information

Finding Complex Solutions of Quadratic Equations

Finding Complex Solutions of Quadratic Equations y - y - - - x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes

More information

10.1 Inverses of Simple Quadratic and Cubic Functions

10.1 Inverses of Simple Quadratic and Cubic Functions COMMON CORE Locker LESSON 0. Inverses of Simple Quadratic and Cubic Functions Name Class Date 0. Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of

More information

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a

More information

3.1 Solving Quadratic Equations by Taking Square Roots

3.1 Solving Quadratic Equations by Taking Square Roots COMMON CORE -8-16 1 1 10 8 6 0 y Locker LESSON.1 Solving Quadratic Equations by Taking Square Roots Name Class Date.1 Solving Quadratic Equations by Taking Square Roots Essential Question: What is an imaginary

More information

Looking Ahead to Chapter 10

Looking Ahead to Chapter 10 Looking Ahead to Chapter Focus In Chapter, you will learn about polynomials, including how to add, subtract, multiply, and divide polynomials. You will also learn about polynomial and rational functions.

More information

A repeated root is a root that occurs more than once in a polynomial function.

A repeated root is a root that occurs more than once in a polynomial function. Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows

More information

Just DOS Difference of Perfect Squares. Now the directions say solve or find the real number solutions :

Just DOS Difference of Perfect Squares. Now the directions say solve or find the real number solutions : 5.4 FACTORING AND SOLVING POLYNOMIAL EQUATIONS To help you with #1-1 THESE BINOMIALS ARE EITHER GCF, DOS, OR BOTH!!!! Just GCF Just DOS Difference of Perfect Squares Both 1. Break each piece down.. Pull

More information

6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property

6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property 6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.

More information

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5 Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There

More information

Using Properties of Exponents

Using Properties of Exponents 6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve real-life problems. VOCABULARY

More information

Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition

Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition Students have the options of either purchasing the loose-leaf

More information

Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2 6-5 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Factor completely. 1. 2y 3 + 4y 2 30y 2y(y 3)(y + 5) 2. 3x 4 6x 2 24 Solve each equation. 3(x 2)(x + 2)(x 2 + 2) 3. x 2 9 = 0 x = 3, 3 4. x 3 + 3x

More information

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions

More information

Domain, Range, and End Behavior

Domain, Range, and End Behavior Locker LESSON 1.1 Domain, Range, and End Behavior Common Core Math Standards The student is epected to: F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship

More information

Chapter 9: Roots and Irrational Numbers

Chapter 9: Roots and Irrational Numbers Chapter 9: Roots and Irrational Numbers Index: A: Square Roots B: Irrational Numbers C: Square Root Functions & Shifting D: Finding Zeros by Completing the Square E: The Quadratic Formula F: Quadratic

More information

13.2 Exponential Decay Functions

13.2 Exponential Decay Functions 6 6 - - Locker LESSON. Eponential Deca Functions Common Core Math Standards The student is epected to: F.BF. Identif the effect on the graph of replacing f() b f() + k, kf(), f(k), and f( + k) for specific

More information

Lesson 5b Solving Quadratic Equations

Lesson 5b Solving Quadratic Equations Lesson 5b Solving Quadratic Equations In this lesson, we will continue our work with Quadratics in this lesson and will learn several methods for solving quadratic equations. The first section will introduce

More information

Polynomials and Polynomial Functions

Polynomials and Polynomial Functions Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial

More information

Solving Quadratic Equations Review

Solving Quadratic Equations Review Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic

More information

Explore 1 Graphing and Analyzing f(x) = e x. The following table represents the function ƒ (x) = (1 + 1 x) x for several values of x.

Explore 1 Graphing and Analyzing f(x) = e x. The following table represents the function ƒ (x) = (1 + 1 x) x for several values of x. 1_ 8 6 8 Locker LESSON 13. The Base e Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes of the graphs of ƒ () = b and ƒ () = log b () where b is, 1, and e

More information

Name: 6.4 Polynomial Functions. Polynomial in One Variable

Name: 6.4 Polynomial Functions. Polynomial in One Variable Name: 6.4 Polynomial Functions Polynomial Functions: The expression 3r 2 3r + 1 is a in one variable since it only contains variable, r. KEY CONCEPT Polynomial in One Variable Words A polynomial of degree

More information

Solving Polynomial Equations 3.5. Essential Question How can you determine whether a polynomial equation has a repeated solution?

Solving Polynomial Equations 3.5. Essential Question How can you determine whether a polynomial equation has a repeated solution? 3. Solving Polynomial Equations Essential Question Essential Question How can you determine whether a polynomial equation has a repeated solution? Cubic Equations and Repeated Solutions USING TOOLS STRATEGICALLY

More information

2.3. Solving Absolute Value Inequalities. Inequalities ENGAGE. 2.3 Solving Absolute Value

2.3. Solving Absolute Value Inequalities. Inequalities ENGAGE. 2.3 Solving Absolute Value Resource Locker LESSO N 2.3 Solving Absolute Value Inequalities Name Class Date 2.3 Solving Absolute Value Inequalities Texas Math Standards The student is expected to: A2.6.F Solve absolute value linear

More information

Scope and Sequence Mathematics Algebra 2 400

Scope and Sequence Mathematics Algebra 2 400 Scope and Sequence Mathematics Algebra 2 400 Description : Students will study real numbers, complex numbers, functions, exponents, logarithms, graphs, variation, systems of equations and inequalities,

More information

Homework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers.

Homework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers. Morgan County School District Re-3 A.P. Calculus August What is the language of algebra? Graphing real numbers. Comparing and ordering real numbers. Finding absolute value. September How do you solve one

More information

Section 4.1: Polynomial Functions and Models

Section 4.1: Polynomial Functions and Models Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial

More information

West Windsor-Plainsboro Regional School District Math A&E Grade 7

West Windsor-Plainsboro Regional School District Math A&E Grade 7 West Windsor-Plainsboro Regional School District Math A&E Grade 7 Page 1 of 24 Unit 1: Introduction to Algebra Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale

More information

CP Algebra 2. Unit 3B: Polynomials. Name: Period:

CP Algebra 2. Unit 3B: Polynomials. Name: Period: CP Algebra 2 Unit 3B: Polynomials Name: Period: Learning Targets 10. I can use the fundamental theorem of algebra to find the expected number of roots. Solving Polynomials 11. I can solve polynomials by

More information

Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A4b & MM2A4c Time allotted for this Lesson: 9 hours

Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A4b & MM2A4c Time allotted for this Lesson: 9 hours Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A4b & MM2A4c Time allotted for this Lesson: 9 hours Essential Question: LESSON 3 Solving Quadratic Equations and Inequalities

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II. 2 nd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II. 2 nd Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

More information

Algebra II Polynomials: Operations and Functions

Algebra II Polynomials: Operations and Functions Slide 1 / 276 Slide 2 / 276 Algebra II Polynomials: Operations and Functions 2014-10-22 www.njctl.org Slide 3 / 276 Table of Contents click on the topic to go to that section Properties of Exponents Review

More information

7.1 Connecting Intercepts and Zeros

7.1 Connecting Intercepts and Zeros Locker LESSON 7. Connecting Intercepts and Zeros Common Core Math Standards The student is epected to: F-IF.7a Graph linear and quadratic functions and show intercepts, maima, and minima. Also A-REI.,

More information

Algebra 1. Mathematics Course Syllabus

Algebra 1. Mathematics Course Syllabus Mathematics Algebra 1 2017 2018 Course Syllabus Prerequisites: Successful completion of Math 8 or Foundations for Algebra Credits: 1.0 Math, Merit The fundamental purpose of this course is to formalize

More information

2.1 Quadratic Functions

2.1 Quadratic Functions Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.

More information

Course Number 432/433 Title Algebra II (A & B) H Grade # of Days 120

Course Number 432/433 Title Algebra II (A & B) H Grade # of Days 120 Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number

More information

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,

More information

3.4 The Fundamental Theorem of Algebra

3.4 The Fundamental Theorem of Algebra 333371_0304.qxp 12/27/06 1:28 PM Page 291 3.4 The Fundamental Theorem of Algebra Section 3.4 The Fundamental Theorem of Algebra 291 The Fundamental Theorem of Algebra You know that an nth-degree polynomial

More information

Lesson 18: Recognizing Equations of Circles

Lesson 18: Recognizing Equations of Circles Student Outcomes Students complete the square in order to write the equation of a circle in center-radius form. Students recognize when a quadratic in xx and yy is the equation for a circle. Lesson Notes

More information

NAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2

NAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2 5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate

More information

23.3. Sampling Distributions. Engage Sampling Distributions. Learning Objective. Math Processes and Practices. Language Objective

23.3. Sampling Distributions. Engage Sampling Distributions. Learning Objective. Math Processes and Practices. Language Objective 23.3 Sampling Distributions Essential Question: How is the mean of a sampling distribution related to the corresponding population mean or population proportion? Explore 1 Developing a Distribution of

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,

More information

SFUSD Mathematics Core Curriculum Development Project

SFUSD Mathematics Core Curriculum Development Project 1 SFUSD Mathematics Core Curriculum Development Project 2014 2015 Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own

More information

Unit 5 Evaluation. Multiple-Choice. Evaluation 05 Second Year Algebra 1 (MTHH ) Name I.D. Number

Unit 5 Evaluation. Multiple-Choice. Evaluation 05 Second Year Algebra 1 (MTHH ) Name I.D. Number Name I.D. Number Unit Evaluation Evaluation 0 Second Year Algebra (MTHH 039 09) This evaluation will cover the lessons in this unit. It is open book, meaning you can use your textbook, syllabus, and other

More information

Absolute Value Inequalities (Advanced Only)

Absolute Value Inequalities (Advanced Only) Unit 1: Linear Functions and Inequalities Time Frame: 5 weeks August 16 to September 20, 2011 Unit Description This unit focuses on the development of concepts of functions that was begun in Algebra I

More information

Math 2 Unit 5: Comparing Functions and Modeling

Math 2 Unit 5: Comparing Functions and Modeling Approximate Time Frame: 2-3 Weeks Connections to Previous Learning: In Math 1, students studied linear and exponential functions and compared them in a modeling context. Students looked at multiple representations

More information

Section 6.6 Evaluating Polynomial Functions

Section 6.6 Evaluating Polynomial Functions Name: Period: Section 6.6 Evaluating Polynomial Functions Objective(s): Use synthetic substitution to evaluate polynomials. Essential Question: Homework: Assignment 6.6. #1 5 in the homework packet. Notes:

More information

Polynomial Functions. Essential Questions. Module Minute. Key Words. CCGPS Advanced Algebra Polynomial Functions

Polynomial Functions. Essential Questions. Module Minute. Key Words. CCGPS Advanced Algebra Polynomial Functions CCGPS Advanced Algebra Polynomial Functions Polynomial Functions Picture yourself riding the space shuttle to the international space station. You will need to calculate your speed so you can make the

More information

Pre-Algebra Lesson Plans

Pre-Algebra Lesson Plans EMS 8 th Grade Math Department Math Florida Standard(s): Learning Goal: Assessments Algebra Preview: Polynomials May 2 nd to June 3 rd, 2016 MAFS.912.A-SSE.1.1b (DOK 2) Interpret expressions that represent

More information

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and

More information

Roots & Zeros of Polynomials. How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.

Roots & Zeros of Polynomials. How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related. Roots & Zeros of Polynomials How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related. A number a is a zero or root of a function y = f (x) if and only if f (a) =

More information

Westmoreland County Public Schools Pacing Guide and Checklist Algebra 1

Westmoreland County Public Schools Pacing Guide and Checklist Algebra 1 Westmoreland County Public Schools Pacing Guide and Checklist 2018-2019 Algebra 1 translate algebraic symbolic quantitative situations represent variable verbal concrete pictorial evaluate orders of ops

More information

Factors, Zeros, and Roots

Factors, Zeros, and Roots Factors, Zeros, and Roots Mathematical Goals Know and apply the Remainder Theorem Know and apply the Rational Root Theorem Know and apply the Factor Theorem Know and apply the Fundamental Theorem of Algebra

More information

California Content Standard. Essentials for Algebra (lesson.exercise) of Test Items. Grade 6 Statistics, Data Analysis, & Probability.

California Content Standard. Essentials for Algebra (lesson.exercise) of Test Items. Grade 6 Statistics, Data Analysis, & Probability. California Content Standard Grade 6 Statistics, Data Analysis, & Probability 1. Students compute & analyze statistical measurements for data sets: 1.1 Compute the mean, median & mode of data sets 1.2 Understand

More information

Use the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (1-2) 4 3 2

Use the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (1-2) 4 3 2 Name: Math 114 Activity 1(Due by EOC Apr. 17) Dear Instructor or Tutor, These problems are designed to let my students show me what they have learned and what they are capable of doing on their own. Please

More information

MATHEMATICS. Perform a series of transformations and/or dilations to a figure. A FAMILY GUIDE FOR STUDENT SUCCESS 17

MATHEMATICS. Perform a series of transformations and/or dilations to a figure. A FAMILY GUIDE FOR STUDENT SUCCESS 17 MATHEMATICS In grade 8, your child will focus on three critical areas. The first is formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a

More information

Albuquerque Public Schools High School District Benchmark Assessment Algebra I Assessment Alignment

Albuquerque Public Schools High School District Benchmark Assessment Algebra I Assessment Alignment NM PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: These standards should be

More information

= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:

= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives: Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations

More information

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide Topic IV: Polynomial Functions, Expressions, and Equations Pacing Date(s) Traditional 18 10/29/18 11/28/18 Block 9 10/29/18 11/28/18 Topic IV Assessment Window 11/16/18 11/28/18 MATHEMATICS FLORIDA STATE

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8

More information

Nine Week SOL Time Allotment. A.4a, b and A.5a - Properties. A.1b and A.3c - Order of Operations. A.1b - Evaluating Expression

Nine Week SOL Time Allotment. A.4a, b and A.5a - Properties. A.1b and A.3c - Order of Operations. A.1b - Evaluating Expression 6/5/2018 Nine Week SOL Time Allotment A.4a, b and A.5a - Properties A.1b and A.3c - Order of Operations A.1b - Evaluating Expression 3 Days 1 Day 4 Days 1 8.17 and 8.18 - Simplifying Expressions 4 Days

More information

MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline

MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 1-75 odd) A. Simplify fractions. B. Change mixed numbers

More information

and Transitional Comprehensive Curriculum. Algebra I Part 2 Unit 7: Polynomials and Factoring

and Transitional Comprehensive Curriculum. Algebra I Part 2 Unit 7: Polynomials and Factoring Algebra I Part Unit 7: Polynomials and Factoring Time Frame: Approximately four weeks Unit Description This unit focuses on the arithmetic operations on polynomial expressions as well as on basic factoring

More information

11.3 Solving Radical Equations

11.3 Solving Radical Equations Name Class Date 11.3 Solving Radical Equations Essential Question: How can you solve equations involving square roots and cube roots? Explore Investigating Solutions of Square Root Equations Resource Locker

More information

3.4 Pascal s Pride. A Solidify Understanding Task

3.4 Pascal s Pride. A Solidify Understanding Task 3.4 Pascal s Pride A Solidify Understanding Task Multiplying polynomials can require a bit of skill in the algebra department, but since polynomials are structured like numbers, multiplication works very

More information

Curriculum Map: Mathematics

Curriculum Map: Mathematics Curriculum Map: Mathematics Course: Honors Algebra II Grade(s): 9/10 Unit 1: Expressions, Equations, and Inequalities In this unit, students review basics concepts and skills of algebra studied in previous

More information

MATH Spring 2010 Topics per Section

MATH Spring 2010 Topics per Section MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line

More information

UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS

UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS This unit investigates quadratic functions. Students study the structure of quadratic expressions and write quadratic expressions in equivalent forms.

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS ALGEBRA II. 1 st Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS ALGEBRA II. 1 st Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS ALGEBRA II 1 st Nine Weeks, 2016-2017 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

More information

Online Courses for High School Students

Online Courses for High School Students Online Courses for High School Students 1-888-972-6237 Algebra I Course Description: Students explore the tools of algebra and learn to identify the structure and properties of the real number system;

More information

13.3 Exponential Decay Functions

13.3 Exponential Decay Functions 6 6 - - Locker LESSON. Eponential Deca Functions Teas Math Standards The student is epected to: A.5.B Formulate eponential and logarithmic equations that model real-world situations, including eponential

More information

Alg 2 Mid Term Review

Alg 2 Mid Term Review Name: Class: Date: ID: A Alg 2 Mid Term Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Solve 4x 2 5x 2 0. A x 5 8 7 8 C x 5 8 7 8 B x 5 8 7 8 i

More information

Algebra 2 Notes AII.7 Polynomials Part 2

Algebra 2 Notes AII.7 Polynomials Part 2 Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Name: Date: Block: Zeros of a Polynomial Function So far: o If we are given a zero (or factor or solution) of a polynomial function, we can use division

More information

Module 1: Whole Numbers Module 2: Fractions Module 3: Decimals and Percent Module 4: Real Numbers and Introduction to Algebra

Module 1: Whole Numbers Module 2: Fractions Module 3: Decimals and Percent Module 4: Real Numbers and Introduction to Algebra Course Title: College Preparatory Mathematics I Prerequisite: Placement with a score below 20 on ACT, below 450 on SAT, or assessing into Basic Applied Mathematics or Basic Algebra using Accuplacer, ASSET

More information

Pre-Calculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and

Pre-Calculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and Pre-Calculus: 1.1 1.2 Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and finding the domain, range, VA, HA, etc.). Name: Date:

More information

Vocabulary Polynomial: A monomial or the sum of two or more monomials whose exponents are positive. Example: 5a 2 + ba 3. 4a b, 1.

Vocabulary Polynomial: A monomial or the sum of two or more monomials whose exponents are positive. Example: 5a 2 + ba 3. 4a b, 1. A.APR.A.1: Arithmetic Operations on Polynomials POLYNOMIALS AND QUADRATICS A.APR.A.1: Arithmetic Operations on Polynomials A. Perform arithmetic operations on polynomials. 1. Understand that polynomials

More information

Exploring Operations Involving Complex Numbers. (3 + 4x) (2 x) = 6 + ( 3x) + +

Exploring Operations Involving Complex Numbers. (3 + 4x) (2 x) = 6 + ( 3x) + + Name Class Date 11.2 Complex Numbers Essential Question: What is a complex number, and how can you add, subtract, and multiply complex numbers? Explore Exploring Operations Involving Complex Numbers In

More information

Course Text. Course Description. Course Objectives. Course Prerequisites. Important Terms. StraighterLine Introductory Algebra

Course Text. Course Description. Course Objectives. Course Prerequisites. Important Terms. StraighterLine Introductory Algebra Introductory Algebra Course Text Dugopolski, Mark. Elementary Algebra, 6th edition. McGraw-Hill, 2009. ISBN 9780077224790 [This text is available as an etextbook at purchase or students may find used,

More information

Mathematics. Algebra Course Syllabus

Mathematics. Algebra Course Syllabus Prerequisites: Successful completion of Math 8 or Foundations for Algebra Credits: 1.0 Math, Merit Mathematics Algebra 1 2018 2019 Course Syllabus Algebra I formalizes and extends the mathematics students

More information

Enhanced Instructional Transition Guide

Enhanced Instructional Transition Guide 1-1 Enhanced Instructional Transition Guide High School Courses Unit Number: 7 /Mathematics Suggested Duration: 9 days Unit 7: Polynomial Functions and Applications (15 days) Possible Lesson 1 (6 days)

More information