Using Properties of Exponents

Size: px
Start display at page:

Download "Using Properties of Exponents"

Transcription

1 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 Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers and let m and n be integers. Product of powers property a m p a n a Power of a power property (a m ) n a Power of a product property (ab) m a b Negative exponent property a m, a 0 Zero exponent property a 0, a 0 Quotient of powers property a m an a, a 0 Power of a quotient property a b m, b 0 Example 1 Evaluating Numerical Expressions 6 a b. 3 2 c. (3 2 ) Algebra 2 Notetaking Guide Chapter 6

2 Example 2 3 a. x 4 y2 Simplifying Algebraic Expressions Power of a quotient property Power of a power property Negative exponent property b. (3a) 3 a 9 a 7 a 9 a 7 Power of a product property Product of powers property Simplify exponent. c. (c c d) d2 c 9 d 2 c 9 d 2 Power of a product property Power of a power property Quotient of powers property Simplify exponents. Zero exponent property Negative exponent property Checkpoint Complete the following exercises. 1. Evaluate (2 2 ) 3 (2 5 jk ) ). 2. Simplify (j ( 1. 2 k) 3 2 Lesson 6.1 Algebra 2 Notetaking Guide 125

3 Example 3 Comparing Real-Life Volumes The radius of a basketball is about 5.7 times greater than the radius of a golf ball. How many times as great as the golf ball s volume is the basketball s volume? Let r represent the radius of the golf ball. Basketball s volume Golf ball s volume 4 3 π πr3 The volume of a sphere is 4 3 πr π 4 3 πr3 Power of a product property The basketball s volume is about golf ball s volume. Quotient of powers property Zero exponent property Approximate power. times as great as the Example 4 Using Scientific Notation in Real Life Greenland covers about square kilometers and has approximately people. About how many square kilometers are there per person? Homework Land area P opulation There are about Divide land area by population. Quotient of powers property Use a calculator. Write in standard notation. square kilometers per person. 126 Algebra 2 Notetaking Guide Chapter 6

4 6.2 Evaluating and Graphing Polynomial Functions Goals p Evaluate polynomial functions. p Graph polynomial functions. Your Notes VOCABULARY Polynomial function Leading coefficient Constant term Degree Standard form of a polynomial function End behavior Example 1 Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type, and leading coefficient. a. f(x) 2x 1.2 2x 3 4x 2 b. f(x) 0.32x x 3 71 a. The function a polynomial function because does not have a. b. The function a polynomial function. Its standard form is. It has degree, so it is a function. The leading coefficient is. Lesson 6.2 Algebra 2 Notetaking Guide 127

5 Checkpoint Complete the following exercise. 1. State the degree, type, and leading coefficient of f(x) x 4 2x 3 4x 2 x 5. Example 2 Using Synthetic Substitution Use synthetic substitution to evaluate f(x) 2x 4 4x 3 x 2 1 when x 2. Write the value of x and the coefficients of f(x) as shown. Bring down the leading coefficient. Multiply by and write the result in the next column. the numbers in that column and write the sum below the line. Continue to multiply and add. x-value 2x 4 (4x 3 ) x 2 0x 1 Polynomial in standard form Coefficients f(2) END BEHAVIOR FOR POLYNOMIAL FUNCTIONS The graph of f(x) a n x n a n 1 x n 1... a 1 x a 0 has this end behavior: For a n > 0 and n even, f(x) as x and f(x) as x. For a n > 0 and n odd, f(x) f(x) as x. For a n < 0 and n even, f(x) f(x) as x. For a n < 0 and n odd, f(x) f(x) as x. as x and as x and as x and 128 Algebra 2 Notetaking Guide Chapter 6

6 Example 3 Graphing Polynomial Functions Graph (a) f(x) x 3 2x 2 x 1 and (b) f(x) x 4 4x 2 x 1. a. Make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. x f (x) x 1 The degree is and the leading coefficient is, so f(x) as x and f(x) as x. b. Make a table of values and y plot the corresponding points. 3 Connect the points with a smooth curve and check the end behavior y x x 1 f (x) The degree is and the 5 leading coefficient is, so f(x) as x and f(x) as x. 3 Checkpoint Complete the following exercises using the function f(x) x 4 x 3 4x 2 4x 3. Homework 2. Evaluate f(x) for x 4 3. Graph f(x). using synthetic substitution. y x 2 6 Lesson 6.2 Algebra 2 Notetaking Guide 129

7 6.3 Adding, Subtracting, and Multiplying Polynomials Goals p Add, subtract, and multiply polynomials. p Use polynomial operations in real-life problems. Your Notes Example 1 Adding Polynomials Vertically and Horizontally a. 2x 3 3x 2 7x 5 3x 3 2x 2 3x b. (x 3 6x 4) (6x 3 2x 2 9x 2) Example 2 Subtracting Polynomials Vertically and Horizontally a. 7x 3 4x 2 x 4 7x 3 4x 2 x 4 (5x 3 x 2 x 6) b. (3x 4 9x 3 2) (2x 4 6x 3 2x 4) 3x 4 9x 3 2 Example 3 Multiplying Polynomials Vertically x 2 x 3 x 2 Multiply x 2 x 3 by 2. Multiply x 2 x 3 by x. Combine like terms. 130 Algebra 2 Notetaking Guide Chapter 6

8 Example 4 Multiplying Polynomials Horizontally Multiply the polynomials. (x 2)(3x 2 x 5) (x 2) (x 2) (x 2) Example 5 Multiplying Three Binomials Multiply the polynomials. (x 2)(x 1)(x 3) ( )(x 3) ( )(x) ( )(3) SPECIAL PRODUCT PATTERNS Sum and Difference Example (a b)(a b) a 2 b 2 (x 3)(x 3) Square of a Binomial (a b) 2 a 2 2ab b 2 (y 4) 2 (a b) 2 a 2 2ab b 2 (3t 2 2) 2 Cube of a Binomial (a b) 3 (x 1) 3 a 3 3a 2 b 3ab 2 b 3 (a b) 3 (p 2) 3 a 3 3a 2 b 3ab 2 b 3 Lesson 6.3 Algebra 2 Notetaking Guide 131

9 Example 6 Using Special Product Patterns Multiply the polynomials. a. (3z 4)(3z 4) ( ) 2 2 b. (4x 2 3y) 2 ( ) 2 2( )( ) ( ) 2 c. (cd 4) 3 ( ) 3 3( ) 2 ( ) 3( )( ) 2 3 Checkpoint Complete the following exercises. 1. Add (3x 2 3x 7) (x 2 9x 5). 2. Subtract (6x 3 x 2 1) (2x 3 6x 3). 3. Multiply (x 3)(x 1)(x 7). Homework 4. Multiply (2w 3) Algebra 2 Notetaking Guide Chapter 6

10 6.4 Factoring and Solving Polynomial Equations Goals p Factor polynomial expressions. p Use factoring to solve polynomial equations. Your Notes VOCABULARY Factor by grouping Quadratic form SPECIAL FACTORING PATTERNS Sum of Two Cubes a 3 b 3 (a b)(a 2 ab b 2 ) Example x 3 8 (x 2)( ) Difference of Two Cubes a 3 b 3 (a b)(a 2 ab b 2 ) Example 8x 3 1 (2x 1)( ) Example 1 Factoring the Sum or Difference of Cubes Factor each polynomial. a. x 3 64 x 3 3 (x )( ) b. 54y 4 16y 2y( ) 2y ( )3 3 2y( )( ) Lesson 6.4 Algebra 2 Notetaking Guide 133

11 Example 2 Factoring by Grouping Factor the polynomial x 3 3x 2 4x 12. x 3 3x 2 4x 12 x 2 ( ) 4( ) Factor by grouping. Difference of squares Example 3 Factoring Polynomials in Quadratic Form Factor (a) 16x 4 1 and (b) 2x 6 10x 4 12x 2. a. 16x 4 1 ( ) 2 2 b. 2x 6 10x 4 12x 2 2x 2 ( ) Example 4 Solving a Polynomial Equation Solve x 4 4 5x 2. x 4 4 5x Write original equation. Rewrite in standard form. Factor trinomial. 0 Factor difference of squares. x, x, x, or x Zero product property The solutions are. Check these in the original equation. 134 Algebra 2 Notetaking Guide Chapter 6

12 Checkpoint Factor each polynomial in Exercises x x 3 x 2 2x 2 3. x 4 7x Solve x 5 2x x 3. Example 5 Solving a Polynomial Equation in Real Life A rectangular swimming pool has a volume of 512 cubic feet. The pool s dimensions are x feet deep by 6x 8 feet long by 6x 16 feet wide. How deep is the pool? Verbal Model Volume p Length p Width Labels Volume (cubic feet) Depth (feet) Length (feet) Width (feet) Homework Algebraic Model Standard form 0 Factor by grouping. 0 The only real solution is x, so 6x 8 and 6x 16. The pool is feet deep. The dimensions are. Lesson 6.4 Algebra 2 Notetaking Guide 135

13 6.5 The Remainder and Factor Theorems Goals p Divide polynomials and relate the result to the remainder theorem and the factor theorem. p Use polynomial division in real-life problems. Your Notes VOCABULARY Polynomial long division Synthetic division Example 1 Using Polynomial Long Division Divide 4x 4 x 2 18x 8 by x 2 2x 3. Write division in the same format you would use when dividing numbers. Include a 0 as the coefficient of x 3. x 2 2x 34x 4 0x 3 12x 2 18x 8 Write the result as follows. 4x 4 x 2 18x 8 x 2 2x Algebra 2 Notetaking Guide Chapter 6

14 REMAINDER THEOREM If a polynomial f(x) is divided by x k, the remainder is r. Example 2 Using Synthetic Division Divide x 3 x 2 5x 3 by x 2. To find the value of k, rewrite the divisor in the form x k. Because x 2 x, k x 3 x 2 5x 3 x 2 FACTOR THEOREM A polynomial f(x) has a factor x k if and only if f(k). Example 3 Factoring a Polynomial Factor f(x) x 3 19x 30 given that f(5) 0. Because f(5) 0, you know that is a factor of f(x). Use synthetic division to find the other factors The result gives the coefficients of the quotient. x 3 19x 30 ( )( ) ( )( )( ) Lesson 6.5 Algebra 2 Notetaking Guide 137

15 Example 4 Finding Zeros of a Polynomial Function A zero of f(x) x 3 x 2 4x 4 is x 1. Find the other zeros. Because f(1) 0, you know that is a factor of f(x). Use synthetic division to find the other factors The result gives the coefficients of the quotient. f(x) x 3 x 2 4x 4 ( )( ) ( )( )( ) By the factor theorem, the zeros of f are. Checkpoint Complete the following exercises. 1. Use long division to divide x 2 4x 1 by x Use synthetic division to divide 2x 3 x 2 3x 4 by x 1. Homework 3. Factor f(x) 2x 3 x 2 25x 12 given that f(4) A zero of f(x) x 4 5x 2 4 is 1. Find the other zeros. 138 Algebra 2 Notetaking Guide Chapter 6

16 6.6 Finding Rational Zeros Goals p Find the rational zeros of a polynomial function. p Use polynomial equations to solve real-life problems. THE RATIONAL ZERO THEOREM If f(x) a n x n... a 1 x a 0 has coefficients, then every rational zero of f has the following form: p q factor of constant term factor of leading coefficient Example 1 Using the Rational Zero Theorem Find the rational zeros of f(x) x 3 5x 2 2x 8. List the possible rational zeros. The leading coefficient is and the constant term is. So, the possible rational zeros are: x,,, Test these zeros using synthetic division. Test x is a zero. Because is a zero of f, write f(x). Factor the trinomial and use the factor theorem. f(x) The zeros of f are. Lesson 6.6 Algebra 2 Notetaking Guide 139

17 Example 2 Using the Rational Zero Theorem Find all real zeros of f(x) 6x 4 7x 3 19x 2 5x 6. List the possible rational zeros of f:. Choose values to check by using your graphing utility to graph the function. Two reasonable choices are x and x. Check the value using synthetic division is a zero. Factor out a binomial using the result of the synthetic division. f(x) Rewrite as product of factors. Factor from the second factor. Multiply the first factor by. Repeat the steps above for g(x). Any zero of g will also be a zero of f. The possible rational zeros of g are x. Confirm that the value x is a zero by using synthetic division. f(x) Find the remaining zeros of f by using the quadratic formula to solve. The real zeros of f are,,, and. 140 Algebra 2 Notetaking Guide Chapter 6

18 Checkpoint Complete the following exercise. 1. Find all real zeros of f(x) 5x 4 6x 3 24x 2 15x 2. Example 3 Writing and Using a Polynomial Model You are making a wooden rectangular box. You want the volume of the box to be 135 cubic inches. You want the length of each side of the square base to be x inches and the height to be x 12 inches. What are the dimensions? The volume is V Bh where B base area and h height. Verbal Model Labels Volume Volume Area of Base Height p (cubic inches) (square inches) (inches) Height Algebraic Model The possible rational solutions are x,,,,,,, and. Homework In this case, it makes sense to test only positive x-values. 1 3 So, x is a solution. The base should be inches by inches. The height should be inches. Lesson 6.6 Algebra 2 Notetaking Guide 141

19 6.7 Using the Fundamental Theorem of Algebra Goals p Use the fundamental theorem of algebra. p Use technology to approximate zeros. Your Notes VOCABULARY Repeated solution For the equation f(x) 0, k is a repeated solution if and only if the factor (x k) has degree greater than 1 when f is factored completely. Example 1 Finding the Number of s or Zeros a. The equation x 3 2x 2 x 2 0 has three solutions: 2, i, i. b. The function f(x) x 4 3x 3 4x 2 has four zeros: 0, 0, 1, 4. Example 2 Finding the Zeros of a Polynomial Function Find all the zeros of f(x) x 5 2x 4 3x 3 6x 2 4x 8. The possible rational zeros are 1, 2, 4, and 8. Using synthetic division, you can determine that 2 is a repeated zero and that 2 is also a zero. You can write the function in factored form as follows: f(x) (x 2)(x 2)(x 2)(x 2 1). Complete the factorization. f(x) (x 2)(x 2)(x 2)(x i)(x i) The five zeros are 2, 2, 2, i, and i. The graph of f is shown at the right. Note that only the real zeros appear as x-intercepts. Also note that the graph only touches the x-axis at the repeated zero x 2, but crosses the x-axis at the zero x x 5 y 142 Algebra 2 Notetaking Guide Chapter 6

20 Checkpoint Complete the following exercises. 1. State the number of zeros of f(x) x 3 3x 2 5x 25 and tell what they are. 2. Find all zeros of f(x) x 4 7x 2 18x 10. Example 3 Using Zeros to Write Polynomial Functions Write a polynomial function f of least degree that has real coefficients, leading coefficient 1, and zeros 1 and 2 i. Because the coefficients are real and 2 i is a zero, must also be a zero. Use the three zeros and the factor theorem to write f(x) as a product of three factors. f(x) ( )[x ( )][x ( )] Factored form ( )[ ][ ] Regroup. Multiply. Expand, use i 2. Simplify. Multiply. Combine like terms. Check You can check by evaluating f(x) at each of its zeros. Lesson 6.7 Algebra 2 Notetaking Guide 143

21 Checkpoint Complete the following exercise. 3. Write a polynomial function of least degree that has real coefficients, a leading coefficient of 1, and 4, 3i, and 3i as zeros. Example 4 Approximating Real Zeros Approximate the real zeros of f(x) x 4 5x 3 6x 2 20x 8. Use a graphing calculator to approximate the real zeros of the function. Use the Zero (or Root) feature. You can see that the real zeros are about. The polynomial function has degree, so there must be other zeros. These may be repeats of the real zeros or imaginary. In this case, they are : x. Homework Checkpoint Complete the following exercise. 4. Approximate the real zeros of f(x) x 5 6x 4 10x 3 18x 2 21x. 144 Algebra 2 Notetaking Guide Chapter 6

22 6.8 Analyzing Graphs of Polynomial Functions Goals p Analyze the graph of a polynomial function. p Use polynomial functions in real life. Your Notes VOCABULARY Local maximum Local minimum ZEROS, FACTORS, SOLUTIONS, AND INTERCEPTS Let f(x) a n x n a n 1 x n 1... a 1 x a 0 be a polynomial function. The following statements are equivalent. Zero: is a zero of the polynomial function f. Factor: is a factor of the polynomial f(x). : is a solution of the polynomial equation f(x) 0. If k is a real number, then the following is also equivalent. x-intercept: is an x-intercept of the graph of the polynomial function f. Lesson 6.8 Algebra 2 Notetaking Guide 145

23 Example 1 Using x-intercepts to Graph a Polynomial Function Graph the function f(x) 1 2 (x 1)2 (x 3). Plot x-intercepts. Because x 1 and x 3 are factors of f(x), and are the x-intercepts of the graph of f. Plot the points (, ) and (, ). Plot points between and beyond the x-intercepts. x y Determine the end behavior of the graph. Because f(x) has linear factors of the form x k and a constant factor of, it is a function with a leading coefficient. Therefore, f(x) f(x) as x. Draw the graph so that it passes through the points you plotted and has the appropriate end behavior. as x and y x 1 TURNING POINTS OF POLYNOMIAL FUNCTIONS The graph of every polynomial function of degree n has at most turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly turning points. 146 Algebra 2 Notetaking Guide Chapter 6

24 Example 2 Finding Turning Points Graph each function. Identify the x-intercepts and the points where the local maximums and local minimums occur. a. f(x) x 3 4x 2 x 4 b. f(x) x 4 2x 3 2x 2 3x a. Use a graphing calculator to graph the function. Notice that the graph has x-intercepts and turning points. You can use the graphing calculator s Zero, Maximum, and Minimum features to approximate the coordinates of the points. The x-intercepts of the graph are. The function has a local minimum at (, ) and a local maximum at (, ). b. Use a graphing calculator to graph the function. Notice that the graph has x-intercepts and turning points. You can use the graphing calculator s Zero, Maximum, and Minimum features to approximate the coordinates of the points. The x-intercepts of the graph are. The function has local maximums at (, ) and (, ), and it has a local minimum at (, ). Lesson 6.8 Algebra 2 Notetaking Guide 147

25 Checkpoint Complete the following exercises. 1. Graph f(x) (x 1)(3x 1)(2x 5). 6 y x 6 2. Use a graphing calculator to identify the x-intercepts, local maximums, and local minimums of the graph of f(x) x 4 x 3 6x 2 4x 2. Homework 148 Algebra 2 Notetaking Guide Chapter 6

26 6.9 Modeling with Polynomial Functions Goals p Use finite differences. p Use technology to find polynomial models. Your Notes VOCABULARY Finite differences Example 1 Writing a Cubic Function Write the cubic function whose graph is shown at the right. Use the three given x-intercepts to write f(x) a. To find a, substitute the coordinates of the fourth point. (1, 0) 1 (2, 0) a, so a. f(x) x (0, 4) Check Check the graph s end behavior. The degree of f is and a 0, so f(x) as x and f(x) as x. 5 y (2, 0) Lesson 6.9 Algebra 2 Notetaking Guide 149

27 Checkpoint Complete the following exercise. 1. Write the cubic function of the graph shown. (0, 4) y 1 (2, 0) 3 (1, 0) (1, 0) x Example 2 Finding Finite Differences An equation for a polynomial function is f(n) n 3 2n 2 3n 1. Show that this function has constant third-order differences. Write the first several function values. Find the first-order differences by subtracting consecutive function values. Then find the second-order differences by subtracting consecutive differences. Finally, find the third-order differences by subtracting consecutive differences. f(1) f(2) f(3) f(4) f(5) f(6) Function values for equally-spaced n-values First-order differences Second-order differences Third-order differences PROPERTIES OF FINITE DIFFERENCES 1. If a polynomial function f(x) has degree n, then the nth-order differences of function values for equally spaced x-values are. 2. Conversely, if the nth-order differences of equally-spaced data are, then the data can be represented by a polynomial function of degree n. 150 Algebra 2 Notetaking Guide Chapter 6

28 Example 3 Modeling with Finite Differences The values of a polynomial function for five consecutive whole numbers are given below. Write a polynomial function for f(n). f(1) 5, f(2) 14, f(3) 27, f(4) 44, and f(5) 65 Begin by finding the finite differences. f(1) f(2) f(3) f(4) f(5) Function values for equally-spaced n-values First-order differences Second-order differences Because the differences are constant, you know that the numbers can be represented by a function which has the form f(n). By substituting the first three values into the function, you can obtain a system of three linear equations in variables. Using a calculator to solve the system gives a, b, and c. The polynomial function is f(n). Checkpoint Complete the following exercise. Homework 2. Values of a polynomial function for six consecutive whole numbers are given. Write a polynomial function for f(n). f(1) 2, f(2) 8, f(3) 22, f(4) 50, f(5) 98, and f(6) 172 Lesson 6.9 Algebra 2 Notetaking Guide 151

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

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

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division. Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10

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

Power and Polynomial Functions. College Algebra

Power and Polynomial Functions. College Algebra Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.

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

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

Chapter 2 notes from powerpoints

Chapter 2 notes from powerpoints Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational

More information

Lesson 2: Introduction to Variables

Lesson 2: Introduction to Variables Lesson 2: Introduction to Variables Topics and Objectives: Evaluating Algebraic Expressions Some Vocabulary o Variable o Term o Coefficient o Constant o Factor Like Terms o Identifying Like Terms o Combining

More information

6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4

6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4 2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using

More information

2.1 Evaluate and Graph Polynomial

2.1 Evaluate and Graph Polynomial 2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of

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

Subtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know.

Subtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know. REVIEW EXAMPLES 1) Solve 9x + 16 = 0 for x. 9x + 16 = 0 9x = 16 Original equation. Subtract 16 from both sides. 16 x 9 Divide both sides by 9. 16 x Take the square root of both sides. 9 4 x i 3 Evaluate.

More information

Unit 1: Polynomial Functions SuggestedTime:14 hours

Unit 1: Polynomial Functions SuggestedTime:14 hours Unit 1: Polynomial Functions SuggestedTime:14 hours (Chapter 3 of the text) Prerequisite Skills Do the following: #1,3,4,5, 6a)c)d)f), 7a)b)c),8a)b), 9 Polynomial Functions A polynomial function is an

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

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions

More information

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power. MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world

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

Skills Practice Skills Practice for Lesson 10.1

Skills Practice Skills Practice for Lesson 10.1 Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).

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

3.4 Solve Equations with Variables

3.4 Solve Equations with Variables 3.4 Solve Equations with Variables on Both Sides Goal p Solve equations with variables on both sides. Your Notes VOCABULARY Identity Example 1 Solve 15 1 4a 5 9a 2 5. Solve an equation with variables on

More information

NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.

NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer. 2-1 Power and Radical Functions What You ll Learn Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 2-1 Active Vocabulary extraneous solution

More information

Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra 2. Unit 9. Polynomials Name Period Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

More information

Dividing Polynomials

Dividing Polynomials 3-3 3-3 Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Divide using long division. 1. 161 7 2. 12.18 2.1 23 5.8 Divide. 3. 4. 6x + 15y 3 7a 2 ab a 2x + 5y 7a b Objective

More information

NC Math 3 Modelling with Polynomials

NC Math 3 Modelling with Polynomials NC Math 3 Modelling with Polynomials Introduction to Polynomials; Polynomial Graphs and Key Features Polynomial Vocabulary Review Expression: Equation: Terms: o Monomial, Binomial, Trinomial, Polynomial

More information

Ch 7 Summary - POLYNOMIAL FUNCTIONS

Ch 7 Summary - POLYNOMIAL FUNCTIONS Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)

More information

PreCalculus: Semester 1 Final Exam Review

PreCalculus: Semester 1 Final Exam Review Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain

More information

24. Find, describe, and correct the error below in determining the sum of the expressions:

24. Find, describe, and correct the error below in determining the sum of the expressions: SECONDARY 3 HONORS ~ Unit 2A Assignments SECTION 2.2 (page 69): Simplify each expression: 7. 8. 9. 10. 11. Given the binomials and, how would you find the product? 13. Is the product of two polynomials

More information

More Polynomial Equations Section 6.4

More Polynomial Equations Section 6.4 MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division

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

Chapter 3 Polynomial Functions

Chapter 3 Polynomial Functions Trig / Coll. Alg. Name: Chapter 3 Polynomial Functions 3.1 Quadratic Functions (not on this test) For each parabola, give the vertex, intercepts (x- and y-), axis of symmetry, and sketch the graph. 1.

More information

Dividing Polynomials

Dividing Polynomials 5.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.C Dividing Polynomials Essential Question How can you use the factors of a cubic polynomial to solve a division problem involving the polynomial? Dividing

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

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

Grade 12 Pre-Calculus Mathematics Notebook. Chapter 3. Polynomial Functions

Grade 12 Pre-Calculus Mathematics Notebook. Chapter 3. Polynomial Functions Grade 1 Pre-Calculus Mathematics Notebook Chapter 3 Polynomial Functions Outcomes: R11 & R1 3.1 Characteristics of Polynomial Functions R1 (p.106-113) Polynomial Function = a function of the form where

More information

Unit 2 Polynomial Expressions and Functions Note Package. Name:

Unit 2 Polynomial Expressions and Functions Note Package. Name: MAT40S Mr. Morris Unit 2 Polynomial Expressions and Functions Note Package Lesson Homework 1: Long and Synthetic p. 7 #3 9, 12 13 Division 2: Remainder and Factor p. 20 #3 12, 15 Theorem 3: Graphing Polynomials

More information

Controlling the Population

Controlling the Population Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1

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

Polynomial and Synthetic Division

Polynomial and Synthetic Division Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1

More information

Identify polynomial functions

Identify polynomial functions EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. a. h (x) = x 4 1 x 2

More information

Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks

Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks Chapter/ Lesson Common Core Standard(s) 1-1 SMP1 1. How do you use a number line to graph and order real numbers? 2. How do you identify

More information

The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function

The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function 8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line

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

Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x ) A) x = 5 B) x = -6 C) x = -5 D) x = 6

Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x ) A) x = 5 B) x = -6 C) x = -5 D) x = 6 Review for Final Exam Math 124A (Flatley) Name Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x - 14 1) A) x = 5 B) x = -6 C) x = -5 D) x = 6 Solve the linear equation.

More information

Polynomial Functions

Polynomial Functions Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary),

More information

Using the Laws of Exponents to Simplify Rational Exponents

Using the Laws of Exponents to Simplify Rational Exponents 6. Explain Radicals and Rational Exponents - Notes Main Ideas/ Questions Essential Question: How do you simplify expressions with rational exponents? Notes/Examples What You Will Learn Evaluate and simplify

More information

4.6 Model Direct Variation

4.6 Model Direct Variation 4.6 Model Direct Variation Goal p Write and graph direct variation equations. Your Notes VOCABULARY Direct variation Constant of variation Eample Identif direct variation equations Tell whether the equation

More information

Solve for the variable by transforming equations:

Solve for the variable by transforming equations: Cantwell Sacred Heart of Mary High School Math Department Study Guide for the Algebra 1 (or higher) Placement Test Name: Date: School: Solve for the variable by transforming equations: 1. y + 3 = 9. 1

More information

Common Core Algebra 2 Review Session 1

Common Core Algebra 2 Review Session 1 Common Core Algebra 2 Review Session 1 NAME Date 1. Which of the following is algebraically equivalent to the sum of 4x 2 8x + 7 and 3x 2 2x 5? (1) 7x 2 10x + 2 (2) 7x 2 6x 12 (3) 7x 4 10x 2 + 2 (4) 12x

More information

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.3 Real Zeros of Polynomial Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Use long

More information

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

Algebra 2 Summer Math Answer Section

Algebra 2 Summer Math Answer Section Algebra 2 Summer Math Answer Section 1. ANS: A PTS: 1 DIF: Level B REF: MALG0064 STA: SC.HSCS.MTH.00.AL1.A1.I.C.4 TOP: Lesson 1.1 Evaluate Expressions KEY: word volume cube area solid 2. ANS: C PTS: 1

More information

Chapter 2 Notes: Polynomials and Polynomial Functions

Chapter 2 Notes: Polynomials and Polynomial Functions 39 Algebra 2 Honors Chapter 2 Notes: Polynomials and Polynomial Functions Section 2.1: Use Properties of Exponents Evaluate each expression (3 4 ) 2 ( 5 8 ) 3 ( 2) 3 ( 2) 9 ( a2 3 ( y 2 ) 5 y 2 y 12 rs

More information

Algebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions

Algebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions OpenStax-CNX module: m49349 1 Zeros of Polynomial Functions OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will:

More information

Math 3 Variable Manipulation Part 3 Polynomials A

Math 3 Variable Manipulation Part 3 Polynomials A Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does

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

a real number, a variable, or a product of a real number and one or more variables with whole number exponents a monomial or the sum of monomials

a real number, a variable, or a product of a real number and one or more variables with whole number exponents a monomial or the sum of monomials 5-1 Polynomial Functions Objectives A2.A.APR.A.2 (formerly A-APR.A.3) Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function

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

Completing the Square

Completing the Square 5-7 Completing the Square TEKS FOCUS TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. Additional TEKS

More information

Why? _ v a There are different ways to simplify the expression. one fraction. term by 2a. = _ b 2

Why? _ v a There are different ways to simplify the expression. one fraction. term by 2a. = _ b 2 Dividing Polynomials Then You divided rational expressions. (Lesson 11-5) Now 1Divide a polynomial by a monomial. 2Divide a polynomial by a binomial. Why? The equation below describes the distance d a

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

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t 4 3. 8s 4 + 4s 6 5s 4. 9m + 3 + m 3 + 5m 5. 4 3p 7 3p 4 1 z 1 + 4 6. ( ) 7. 6( x + ) 4 8. 3( h + 4) 3( h

More information

Algebra 2 Honors: Final Exam Review

Algebra 2 Honors: Final Exam Review Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt

More information

Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition. LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in

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

Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.

Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc. Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a

More information

Use Properties of Exponents

Use Properties of Exponents 4. Georgia Performance Standard(s) MMAa Your Notes Use Properties of Eponents Goal p Simplif epressions involving powers. VOCABULARY Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers

More information

MHF4U Unit 2 Polynomial Equation and Inequalities

MHF4U Unit 2 Polynomial Equation and Inequalities MHF4U Unit 2 Polynomial Equation and Inequalities Section Pages Questions Prereq Skills 82-83 # 1ac, 2ace, 3adf, 4, 5, 6ace, 7ac, 8ace, 9ac 2.1 91 93 #1, 2, 3bdf, 4ac, 5, 6, 7ab, 8c, 9ad, 10, 12, 15a,

More information

Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers

More information

Chapter 2 Formulas and Definitions:

Chapter 2 Formulas and Definitions: Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)

More information

Modeling Data. 27 will get new packet. 24 Mixed Practice 3 Binomial Theorem. 23 Fundamental Theorem March 2

Modeling Data. 27 will get new packet. 24 Mixed Practice 3 Binomial Theorem. 23 Fundamental Theorem March 2 Name: Period: Pre-Cal AB: Unit 1: Polynomials Monday Tuesday Block Friday 11/1 1 Unit 1 TEST Function Operations and Finding Inverses 16 17 18/19 0 NO SCHOOL Polynomial Division Roots, Factors, Zeros and

More information

Characteristics of Polynomials and their Graphs

Characteristics of Polynomials and their Graphs Odd Degree Even Unit 5 Higher Order Polynomials Name: Polynomial Vocabulary: Polynomial Characteristics of Polynomials and their Graphs of the polynomial - highest power, determines the total number of

More information

Factors of Polynomials Factoring For Experts

Factors of Polynomials Factoring For Experts Factors of Polynomials SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Discussion Group, Note-taking When you factor a polynomial, you rewrite the original polynomial as a product

More information

LESSON 9.1 ROOTS AND RADICALS

LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical

More information

Math-3. Lesson 3-1 Finding Zeroes of NOT nice 3rd Degree Polynomials

Math-3. Lesson 3-1 Finding Zeroes of NOT nice 3rd Degree Polynomials Math- Lesson - Finding Zeroes of NOT nice rd Degree Polynomials f ( ) 4 5 8 Is this one of the nice rd degree polynomials? a) Sum or difference of two cubes: y 8 5 y 7 b) rd degree with no constant term.

More information

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations

More information

Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.

Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200. Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

More information

Slide 1 / 200. Quadratic Functions

Slide 1 / 200. Quadratic Functions Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

More information

2.4 Multiply Real Numbers

2.4 Multiply Real Numbers 24 Multiply Real Numbers Goal p Multiply real numbers Your Notes VOCABULARY Multiplicative identity THE SIGN OF A PRODUCT The product of two real numbers with the same sign is Examples: 5(2) 5 24(25) 5

More information

9.4 Start Thinking. 9.4 Warm Up. 9.4 Cumulative Review Warm Up. Use a graphing calculator to graph ( )

9.4 Start Thinking. 9.4 Warm Up. 9.4 Cumulative Review Warm Up. Use a graphing calculator to graph ( ) 9.4 Start Thinking Use a graphing calculator to graph ( ) f x = x + 4x 1. Find the minimum of the function using the CALC feature on the graphing calculator. Explain the relationship between the minimum

More information

Name: Class: Date: A. 70 B. 62 C. 38 D. 46

Name: Class: Date: A. 70 B. 62 C. 38 D. 46 Class: Date: Test 2 REVIEW Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Divide: (4x 2 49y 2 ) (2x 7y) A. 2x 7y B. 2x 7y C. 2x 7y D. 2x 7y 2. What is

More information

Unit 7: Rational and Radical Functions Unit Length: 20 days

Unit 7: Rational and Radical Functions Unit Length: 20 days Unit 7: Rational and Radical Functions Unit Length: 20 days Domain: The Real Number System Cluster 1: Extend the properties of exponents to rational exponents. Cluster 2: Use properties of rational and

More information

Multiplication of Polynomials

Multiplication of Polynomials Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is

More information

How many solutions are real? How many solutions are imaginary? What are the solutions? (List below):

How many solutions are real? How many solutions are imaginary? What are the solutions? (List below): 1 Algebra II Chapter 5 Test Review Standards/Goals: F.IF.7.c: I can identify the degree of a polynomial function. F.1.a./A.APR.1.: I can evaluate and simplify polynomial expressions and equations. F.1.b./

More information

7.7. Factoring Special Products. Essential Question How can you recognize and factor special products?

7.7. Factoring Special Products. Essential Question How can you recognize and factor special products? 7.7 Factoring Special Products Essential Question How can you recognize and factor special products? Factoring Special Products LOOKING FOR STRUCTURE To be proficient in math, you need to see complicated

More information

An equation is a statement that states that two expressions are equal. For example:

An equation is a statement that states that two expressions are equal. For example: Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the

More information

Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal

Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Asymptote Example 2: Real-World Example: Use Graphs

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

Chapter 5: Exponents and Polynomials

Chapter 5: Exponents and Polynomials Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5

More information

evaluate functions, expressed in function notation, given one or more elements in their domains

evaluate functions, expressed in function notation, given one or more elements in their domains Describing Linear Functions A.3 Linear functions, equations, and inequalities. The student writes and represents linear functions in multiple ways, with and without technology. The student demonstrates

More information

Ready To Go On? Skills Intervention 7-1 Integer Exponents

Ready To Go On? Skills Intervention 7-1 Integer Exponents 7A Evaluating Expressions with Zero and Negative Exponents Zero Exponent: Any nonzero number raised to the zero power is. 4 0 Ready To Go On? Skills Intervention 7-1 Integer Exponents Negative Exponent:

More information

Polynomial Functions and Models

Polynomial Functions and Models 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

More information

Right Behavior. Left Behavior. Right Behavior

Right Behavior. Left Behavior. Right Behavior U n i t 3 P a r t P a g e 1 Math 3 Unit 3 Part Day 1 Graphing Polynomial Functions Expression 9 x- 3x x + 4x 3 + x + x + 1 5x 4 + x + 10 X 5 + x + 5 3c + 4c /c Type of Function Left Behavior: Right Behavior:

More information

Algebra 2 Segment 1 Lesson Summary Notes

Algebra 2 Segment 1 Lesson Summary Notes Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the

More information

Unit Overview. Content Area: Algebra 2 Unit Title: Preparing for Advanced Algebra Target Course/Grade Level Duration: 10 days

Unit Overview. Content Area: Algebra 2 Unit Title: Preparing for Advanced Algebra Target Course/Grade Level Duration: 10 days Content Area: Algebra 2 Unit Title: Preparing for Advanced Algebra Target Course/Grade Level Duration: 10 days 11 th or 12 th graders Unit Overview Description This chapter 0 contains lessons on topics

More information

Math Analysis Chapter 2 Notes: Polynomial and Rational Functions

Math Analysis Chapter 2 Notes: Polynomial and Rational Functions Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do

More information

Something that can have different values at different times. A variable is usually represented by a letter in algebraic expressions.

Something that can have different values at different times. A variable is usually represented by a letter in algebraic expressions. Lesson Objectives: Students will be able to define, recognize and use the following terms in the context of polynomials: o Constant o Variable o Monomial o Binomial o Trinomial o Polynomial o Numerical

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