CCGPS Frameworks Student Edition. Mathematics. Accelerated CCGPS Analytic Geometry B / Advanced Algebra Unit 6: Polynomial Functions

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1 CCGPS Frameworks Student Edition Mathematics Accelerated CCGPS Analytic Geometry B / Advanced Algebra Unit 6: Polynomial Functions These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

2 TABLE OF CONTENTS Unit 6 Polynomial Functions Overview...3 Standards Addressed in this Unit...3 Enduring Understandings...8 Concepts & Skills to Maintain...8 Selected Terms and Symbols...9 We ve Got to Operate...1 What s Your Identity?...16 Fascinating Fractals...5 Divide and Conquer...36 Factors, Zeros, and Roots: Oh My!...44 Polynomial Patterns...53 July 013 Page of 64

3 OVERVIEW In this unit students will: understand the definition of a polynomial interpret the structure and parts of a polynomial expression including terms, factors, and coefficients simplify polynomial expressions by performing operations, applying the distributive property, and combining like terms use the structure of polynomials to identify ways to rewrite them and write polynomials in equivalent forms to solve problems perform arithmetic operations on polynomials and understand how closure applies under addition, subtraction, and multiplication use Pascal s Triangle to determine coefficients of binomial expansion use polynomial identities to solve problems use complex numbers in polynomial identities and equations derive the formula for the sum of a finite geometric series and use it to solve problems understand and apply the rational Root Theorem understand and apply the Remainder Theorem understand and apply The Fundamental Theorem of Algebra understand the relationship between zeros and factors of polynomials represent, analyze, and solve polynomial functions algebraically and graphically solve systems consisting of a linear equation and a polynomial equation This unit develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the Fundamental Theorem of Algebra. STANDARDS ADDRESSED IN THIS UNIT Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. Interpret the structure of expressions. MCC9 1.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. MCC9 1.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. July 013 Page 3 of 64

4 MCC9 1.A.SSE.1bInterpret complicated expressions by viewing one or more of their parts as a single entity MCC9 1.A.SSE. Use the structure of an expression to identify ways to rewrite it. Write expressions in equivalent forms to solve problems. MCC9 1.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Perform arithmetic operations on polynomials. MCC9 1.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Use polynomial identities to solve problems. MCC9 1.A.APR.4 Prove polynomial identities and use them to describe numerical relationships. MCC9 1.A.APR.5 (+) Know and apply that the Binomial Theorem gives the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) Use complex numbers in polynomial identities and equations. MCC9 1.N.CN.8 (+) Extend polynomial identities to the complex numbers. MCC9 1.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Represent and solve equations and inequalities graphically. MCC9 1.A.REI.11 Explain why the x coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Solve systems of equations. July 013 Page 4 of 64

5 MCC9 1.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic polynomial equation in two variables algebraically and graphically. Understand the relationship between zeros and factors of polynomials. MCC9 1.A.APR. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). MCC9 1.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Analyze functions using different representations. MCC9 1.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. MCC9 1.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important processes and proficiencies with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one s own efficacy). 1. Make sense of problems and persevere in solving them. High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get July 013 Page 5 of 64

6 the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.. Reason abstractly and quantitatively. High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create coherent representations of the problem at hand; consider the units involved; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects. 3. Construct viable arguments and critique the reasoning of others. High school students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. High school students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. High school students learn to determine domains to which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4. Model with mathematics. High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. High school students making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5. Use appropriate tools strategically. High school students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, July 013 Page 6 of 64

7 concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. High school students should be sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6. Attend to precision. High school students try to communicate precisely to others by using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7. Look for and make use of structure. By high school, students look closely to discern a pattern or structure. In the expression x + 9x + 14, older students can see the 14 as 7 and the 9 as + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent expressions, factor and solve equations, and compose functions, and transform figures. 8. Look for and express regularity in repeated reasoning. High school students notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x + x + 1), and (x 1)(x 3 + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content July 013 Page 7 of 64

8 The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics should engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word understand are often especially good opportunities to connect the practices to the content. Students who do not have an understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a missing mathematical knowledge effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential points of intersection between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics. ENDURING UNDERSTANDINGS Viewing an expression as a result of operations on simpler expressions can sometimes clarify its underlying structure. Factoring and other forms of writing polynomials should be explored. The Fundamental Theorem of Algebra is not limited to what can be seen graphically; it applies to real and complex roots. Real and complex roots of higher degree polynomials can be found using the Factor Theorem, Remainder Theorem, Rational Root Theorem, and Fundamental Theorem of Algebra, incorporating complex and radical conjugates. A system of equations is not limited to linear equations; we can find the intersection between a line and a polynomial Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions to the equation. CONCEPTS/SKILLS TO MAINTAIN It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to July 013 Page 8 of 64

9 be spent on conceptual activities that help students develop a deeper understanding of these ideas. Combining like terms and simplifying expressions Long division The distributive property The zero property Properties of exponents Simplifying radicals with positive and negative radicands Factoring quadratic expressions Solving quadratic equations by factoring, taking square roots, using the quadratic formula and utilizing graphing calculator technology to finding zeros/ x-intercepts Observing symmetry, end-behaviors, and turning points (relative maxima and relative minima) on graphs Writing explicit and recursive formulas for geometric sequences SELECT TERMS AND SYMBOLS The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. The websites below are interactive and include a math glossary suitable for high school children. Note At the high school level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks This web site has activities to help students more fully understand and retain new vocabulary. Definitions and activities for these and other terms can be found on the Intermath website. Coefficient: a number multiplied by a variable. Degree: the greatest exponent of its variable End Behavior: the value of f(x) as x approaches positive and negative infinity Fundamental Theorem of Algebra: every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. July 013 Page 9 of 64

10 Geometric Sequence: is a sequence with a constant ratio between successive terms Geometric Series: the expression formed by adding the terms of a geometric sequence Multiplicity: the number of times a root occurs at a given point of a polynomial equation. Pascal s Triangle: an arrangement of the values of n Cr in a triangular pattern where each row corresponds to a value of n Polynomial: a mathematical expression involving a sum of nonnegative integer powers in one or more variables multiplied by coefficients. A polynomial in one variable with n n 1 constant coefficients can be written in an x + an 1 x +... a x + a1x + a0 form. Rational Root Theorem: a theorem that provides a complete list of all possible rational roots of a polynomial equation. It states that every rational zero of the polynomial n n 1 equation f(x) = an x + an 1 x +... a x + a1x + a0,where all coefficients are integers, has p factors of constant term a0 the following form: = q factors of leading coefficient Relative Minimum: a point on the graph where the function is increasing as you move away from the point in the positive and negative direction along the horizontal axis. Relative Maximum: a point on the graph where the function is decreasing as you move away from the point in the positive and negative direction along the horizontal axis. Remainder Theorem: states that the remainder of a polynomial f(x) divided by a linear divisor (x c) is equal to f(c). Roots: solutions to polynomial equations. Sum of a finite geometric series: The sum, S n, of the first n terms of a geometric sequence is given by S n = a 1 a 1 r n = a 1(1 r n ), where a 1 is the first term and r is the 1 r 1 r common ratio (r 1). Sum of an infinite geometric series: The general formula for the sum S of an infinite a1 geometric series a1+ a + a with common ratio r where r < 1 is S =. If an 1 r infinite geometric series has a sum, i.e. if r < 1, then the series is called a convergent geometric series. All other geometric (and arithmetic) series are divergent. July 013 Page 10 of 64 a n

11 Synthetic Division: Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x a). It can be used in place of the standard long division algorithm. Zero: If f(x) is a polynomial function, then the values of x for which f(x) = 0 are called the zeros of the function. Graphically, these are the x intercepts. July 013 Page 11 of 64

12 We ve Got to Operate Previously, you have learned about linear functions, which are first degree polynomial 1 functions that can be written in the form f ( x) = a1x + a0 where a 1 is the slope of the line and a 0 is the y-intercept (Recall: y = mx + b, here m is replaced by a 1 and b is replaced by a 0.) Also, you have learned about quadratic functions, which are nd degree polynomial functions and 1 can be expressed as f ( x) = a x + a1x + a0. These are just two examples of polynomial functions; there are countless others. A polynomial is a mathematical expression involving a sum of nonnegative integer powers in one or more variables multiplied by coefficients. A polynomial in one variable with constant n n 1 coefficients can be written in an x + an 1 x +... a x + a1x + a0 form where a n 0, the exponents are all whole numbers, and the coefficients are all real numbers. 1. What are whole numbers?. What are real numbers? 3. Decide whether each function below is a polynomial. If it is, write the function in standard form. If it is not, explain why. 3 a. f ( x) = x + 5x + 4x + 8 b. f ( x) 1 = x + x c. f ( x) x 3 4 = 5 x + 7x d. f ( x) = x x x e. f ( x) = x g. f ( x) = + 3x x July 013 Page 1 of 64

13 4. Polynomials can be classified by the number terms as well as by the degree of the polynomial. The degree of the polynomial is the same as the term with the highest degree. Complete the following chart. Make up your own expressions for the last three rows. Polynomial Number of Terms Classification Degree Classification f ( x) = monomial constant f ( x) = 3x 1 binomial linear f ( x) = x x + 1 trinomial quadratic f ( x) = 8x + 15 binomial cubic 4 f ( x) = x + 10x + 16 trinomial quartic 5 f ( x) = x monomial quintic 5. To simplify expressions and solve problems, we sometimes need to perform operations with polynomials. We will explore addition and subtraction now. To add or subtract polynomials, just add or subtract the coefficients of like terms. a. Bob owns a small music store. He keeps inventory on his xylophones by using x to represent his professional grade xylophones, x to represent xylophones he sells for recreational use, and constants to represent the number of xylophone instruction manuals he keeps in stock. If the polynomial 5x + x + 4 represents what he has on display in his shop and the polynomial 3x + 6x + 1 represents what he has stocked in the back of his shop, what is the polynomial expression that represents the entire inventory he currently has in stock? b. Suppose a band director makes an order for 6 professional grade xylophones, 13 recreational xylophones and 5 instruction manuals. What polynomial expression would represent Bob s inventory after he processes this order? Explain the meaning of each term. July 013 Page 13 of 64

14 Now find the sum or difference of the following: c. 5x + x 8 d. x x x 7x 1 ( ) x + x+ 1 e. ( 7x 5) ( x 8) + + f. ( a 5a+ 1) + ( a + 3a) g. ( x 5x+ 9) ( 3x + x+ 4) h. ( 5x + xy 7y ) ( 3x 5xy + y ) 6. You have multiplied polynomials previously, but may not have been aware of it. When you utilized the distributive property, you were just multiplying a polynomial by a monomial. Whenever you FOIL, you re just multiplying binomials. In multiplication of polynomials, the central idea is the distributive property. a. An important connection between arithmetic with integers and arithmetic with polynomials can be seen by considering whole numbers in base ten to be polynomials in the base b = 10. Compare the product 47 b + 1b + 3 4b + 7 : 13 with the product ( )( ) 8b 8b b + 4b + 18b b + 1b + 3 4b b b b b. Now compare the product 4 your work! 135 with the product (1b + 3b + 5) ( b + 4). Show July 013 Page 14 of 64

15 7. Find the following products. Be sure to simplify results. a. 3x (x + 8x + 9) b. x (5x x 4) c. ( x 7)( x 5) + d. ( 4x 7)(3x ) e. ( x 3)( x 3x 1) + f. (6x + 4)( x 3x + ) g. ( 4x 7y)( 4x+ 7y) h. ( 3x 4) 3 4 i. ( x 1) j. ( x 1) 8. A set has the closure property under a particular operation if the result of the operation is always an element in the set. If a set has the closure property under a particular operation, then we say that the set is closed under the operation. It is much easier to understand a property by looking at examples than it is by simply talking about it in an abstract way, so let's move on to looking at examples so that you can see exactly what we are talking about when we say that a set has the closure property. a. The set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers. Write a few examples to illustrate this concept: b. The set of integers is not closed under the operation of division because when you divide one integer by another, you don t always get another integer as the answer. Write an example to illustrate this concept: c. Go back and look at all of your answers to problem number 5, in which you added and subtracted polynomials. Do you think that polynomial addition and subtraction is closed? Why or why not? d. Now, go back and look at all of your answers to problems 6 and 7, in which you multiplied polynomials. Do you think that polynomial multiplication is closed? Why or why not? July 013 Page 15 of 64

16 What s Your Identity? Introduction Equivalent algebraic expressions, also called algebraic identities, give us a way to express results with numbers that always work a certain way. In this task you will explore several number tricks that work because of basic algebra rules. You will extend these observations to algebraic expressions in order to prove polynomial identities. Finally, you will learn and apply methods to expand binomials. It is recommended that you do this task with a partner. Materials Pencil Handout Calculator 1. First, you will explore an alternate way to multiply two digit numbers that have the same digit in the ten s place. a. For example, ( 31)(37) can be thought of as (30 + 1)(30 + 7) Using this model, we find the product equals = Verify this solution using a calculator. Are we correct? Rewrite these similarly and use area models to calculate each of the following products: b. (5)(57) c. (16)(13) d. (48)(4) e. (7)(75) July 013 Page 16 of 64

17 . All of the previous products involved addition, how do you think it would be different if they also included subtraction? What if the products involved both addition and subtraction? a. (7)(37) can be thought of as (30-3)(30 + 7) Using this model, we find the product equals = 999. Verify this solution using a calculator. Are we correct? Use both addition and subtraction to rewrite these similarly and use the area models to calculate each of the following products: b. (46)(57) c. (16)(5) d. (38)(4) e. (6)(75) July 013 Page 17 of 64

18 3. Look at problem d above; is there anything special about the binomials that you wrote and the answer that you got? a. With a partner compose three other multiplication questions that use the same idea. Explain your thinking. What must always be true for this special situation to work? Now calculate each of the following using what you have learned about these special binomials. b. (101)(99) c. ()(18) d. (45)(35) e. (.)(1.8) 4. In Question 3, you computed several products of the form ( x y)( x y) product is always of the form x y. + verifying that the a. If we choose values for x and y so that x = y what will the product be? b. Is there any other way to choose numbers to substitute for x and y so that the product x + y x y will equal 0? ( )( ) them? c. In general, if the product of two numbers is zero, what must be true about one of d. These products are called are called conjugates. Give two examples of other conjugates. July 013 Page 18 of 64

19 + = is called a polynomial identity because this statement of equality is true for all values of the variables. e. ( x y)( x y) f. Polynomials in the form of a b are called the difference of two squares. Factor the following using the identity you wrote in problem 4e: x 5 x 11 x x Previously, you ve probably been told you couldn t factor the sum of two squares. These are polynomials that come in the form a + b. Well you can factor these; just not with real numbers. a. Recall 1 = i. What happens when you square both sides? b. Now multiply ( x + 5i)( x 5i) =. Describe what you see. c. I claim that you can factor the sum of two squares just like the difference of two squares, just with i' s after the constant terms. Do you agree? Why or why not? d. This leads us to another polynomial identity for the sum of two squares. a + b = e. Factor the following using the identity you wrote in problem 5d: x + 5 x + 11 x x + 81 July 013 Page 19 of 64

20 6. Now, let s consider another special case to see what happens when the numbers are the same. Start by considering the square below created by adding 4 to the length of each side of a square with side length x. x 4 x x 4 x 4 4 x 16 a. What is the area of the square with side length = x? b. What is the area of the rectangle with length = x and width = 4? c. What is the area of the rectangle with length = 4 and width = x? d. What is the area of the square with side length = 4? e. What is the total area of the square in the model above? x + assuming that x and y are positive numbers. Use your figure to explain the identity for a perfect square trinomial: f. Draw a figure to illustrate the area of a square with side length ( y) ( x + y) = x + xy + y July 013 Page 0 of 64

21 7. This identity gives a rule for squaring a sum. For example, 103 can be written as ( )( ). Use this method to calculate each of the following by making convenient choices for x and y. a. 30 b. 54 c. 65 d Determine the following identity: ( x y) =. Explain or show how you came up with your answer. 9. We will now extend the idea of identities to cubes. a. What is the volume of a cube with side length 4? b. What is the volume of a cube with side length x? c. Now we ll determine the volume of a cube with side length x + 4. First, use the rule for squaring a sum to find the area of the base of the cube: Now use the distributive property to multiply the area of the base by the height, x + 4, and simplify your answer: d. Repeat part 8c for a cube with side length x + y. Write your result as a rule for the cube of a sum. First, use the rule for squaring a sum to find the area of the base of the cube: July 013 Page 1 of 64

22 Now use the distributive property to multiply the area of the base by the height, x + y, and simplify your answer: e. So the identity for a binomial cubed is ( x + y) 3 = f. Determine the following identity: ( x y) 3 =. Explain or show how you came up with your answer. 10. Determine whether the cube of a binomial is equivalent to the sum of two cubes by exploring the following expressions: a. Simplify ( x + ) 3 b. Simplify x c. Is your answer to 10a equivalent to your answer in 10b? d. Simplify ( x + ) ( x x + 4) e. Is your answer to part b equivalent to your answer in part d? f. Your answers to parts b and d should be equivalent. They illustrate two more commonly used polynomial identities: July 013 Page of 64

23 3 3 The Sum of Two Cubes: a + b = ( a + b)( a ab + b ) 3 3 The Difference of Two Cubes: a b = ( a b)( a + ab + b ) g. Simplify the following and describe your results in words: ( x 3)( x + 3x + 9) (x + 5)(4x 10x + 5) 11. Complete the table of polynomial identities to summarize your findings: Description Difference of Two Squares ( a + b)( a b) = Sum of Two Squares ( a + bi)( a bi) = Perfect Square Trinomial ( ) = a + Perfect Square Trinomial ( ) = a b Binomial Cubed ( ) = a b + 3 Binomial Cubed ( ) = a b 3 Sum of Two Cubes a 3 + b 3 = Difference of Two Cubes a 3 b 3 = b Identity 1. Finally, let s look further into how we could raise a binomial to any power of interest. One way would be to use the binomial as a factor and multiply it by itself n times. However, this process could take a long time to complete. Fortunately, there is a quicker way. We will now explore and apply the binomial theorem, using the numbers in Pascal s triangle, to expand a binomial in ( a + b) n form to the th n power. Binomial Expansion Pascal s Triangle th n row ( a + b) 0 1 n = 0 ( a + b) n = 1 ( a + b) 1 1 n = ( a + b) n = 3 a n = 4 ( b) 4 a. Use the fourth row of Pascal s triangle to find the numbers in the fifth row: July 013 Page 3 of 64

24 Use the fifth row of Pascal s triangle to find the numbers in the sixth row: Use the sixth row of Pascal s triangle to find the numbers in the seventh row: b. The binomial coefficients from the third row of Pascal s Triangle are 1, 3, 3, 1, so the expansion of ( x + ) = (1)( x )( ) + (3)( x )( ) + (3)( x )( ) + (1)( x )( ). Describe the pattern you see, and then simplify the result: c. Use Pascal s triangle in order to expand the following: ( x + 5) 3 = ( x + 1) 4 = ( x + 3) 5 = d. To expand binomials representing differences, rather than sums, the binomial coefficients will remain the same but the signs will alternate beginning with positive, then negative, then positive, and so on. Simplify the following and compare the result part b x = (1)( x )( ) (3)( x )( ) + (3)( x )( ) (1)( x )( ( ) ) e. Use Pascal s triangle in order to expand the following: ( x 5) 3 = ( x ) 4 = 5 ( x 10) = July 013 Page 4 of 64

25 FASCINATING FRACTALS LEARNING TASK: Sequences and series arise in many classical mathematics problems as well as in more recently investigated mathematics, such as fractals. The task below investigates some of the interesting patterns that arise when investigating manipulating different figures. Part One: Koch Snowflake 1 (Images obtained from Wikimedia Commons at This shape is called a fractal. Fractals are geometric patterns that are repeated at ever smaller increments. The fractal in this problem is called the Koch snowflake. At each stage, the middle third of each side is replaced with an equilateral triangle. (See the diagram.) To better understand how this fractal is formed, let s create one! On a large piece of paper, construct an equilateral triangle with side lengths of 9 inches. Now, on each side, locate the middle third. (How many inches will this be?) Construct a new equilateral triangle in that spot and erase the original part of the triangle that now forms the base of the new, smaller equilateral triangle. How many sides are there to the snowflake at this point? (Double-check with a partner before continuing.) 1 Often the first picture is called stage 0. For this problem, it is called stage 1. The Sierpinski Triangle, the next problem, presents the initial picture as Stage 0. A number of excellent applets are available on the web for viewing iterations of fractals. July 013 Page 5 of 64

26 Now consider each of the sides of the snowflake. How long is each side? Locate the middle third of each of these sides. How long would one-third of the side be? Construct new equilateral triangles at the middle of each of the sides. How many sides are there to the snowflake now? Note that every side should be the same length. Continue the process a few more times, if time permits. 1. Now complete the first three columns of the following chart. Number of Length of each Perimeter (in) Segments Segment (in) Stage Stage Stage 3. Consider the number of segments in the successive stages. a. Does the sequence of number of segments in each successive stage represent an arithmetic or a geometric sequence (or neither)? Explain. b. What type of graph does this sequence produce? Make a plot of the stage number and number of segments in the figure to help you determine what type of function you will use to model this situation. c. Write a recursive and explicit formula for the number of segments at each stage. d. Find the 7 th term of the sequence. Find 1 th term of the sequence. Now find the 16 th. Do the numbers surprise you? Why or why not? 3. Consider the length of each segment in the successive stages. a. Does this sequence of lengths represent an arithmetic or a geometric sequence (or neither)? Explain. b. Write a recursive and explicit formula for the length of each segment at each stage. c. Find the 7 th term of the sequence. Find the 1 th term of the sequence. Now find the 16 th. July 013 Page 6 of 64

27 d. How is what is happening to these numbers similar or different to what happened to the sequence of the number of segments at each stage? Why are these similarities or differences occurring? 4. Consider the perimeter of the Koch snowflake. a. How did you determine the perimeter for each of the stages in the table? a. Using this idea and your answers in the last two problems, find the approximate perimeters for the Koch snowflake at the 7 th, 1 th, and 16 th stages. b. What do you notice about how the perimeter changes as the stage increases? c. Extension: B. B. Mandelbrot used the ideas above, i.e. the length of segments and the associated perimeters, in his discussion of fractal dimension and to answer the question, How long is the coast of Britain? Research Mandelbrot s argument and explain why some might argue that the coast of Britain is infinitely long. 5. Up to this point, we have not considered the area of the Koch snowflake. Stage a. Using whatever method you know, determine the exact area of the original triangle. b. How do you think we might find the area of the second stage of the snowflake? What about the third stage? The 7 th stage? Are we adding area or subtracting area? c. To help us determine the area of the snowflake, complete the first two columns of the following chart. Note: The sequence of the number of new triangles is represented by a geometric sequence. Consider how the number of segments might help you determine how many new triangles are created at each stage. Number of Segments New triangles created Area of each of the new triangles Total Area of the New Triangles n July 013 Page 7 of 64

28 d. Determine the exact areas of the new triangles and the total area added by their creation for Stages 1 4. Fill in the chart above. (You may need to refer back to problem 1 for the segment lengths.) d. Because we are primarily interested in the total area of the snowflake, let s look at the last column of the table. The values form a sequence. Determine if it is arithmetic or geometric. Then write the recursive and explicit formulas for the total area added by the new triangles at the nth stage. e. Determine how much area would be added at the 10 th stage. 6. Rather than looking at the area at a specific stage, we are more interested in the TOTAL area of the snowflake. So we need to sum the areas. However, these are not necessarily numbers that we want to try to add up. Instead, we can use our rules of exponents and properties of summations to help us find the sum. a. Write an expression using summation notation for the sum of the areas in the snowflake. b. Explain how the expression you wrote in part a is equivalent to 4 n i i= 9 Now, the only part left to determine is how to find the sum of a finite geometric series. Let s take a step back and think about how we form a finite geometric series: S n = a 1 + a 1 r + a 1 r + a 1 r a 1 r n-1 Multiplying both sides by r, we get rs n = a 1 r + a 1 r + a 1 r 3 ++ a 1 r 4 + a 1 r n Subtracting these two equations: S n - rs n = a 1 - a 1 r n Factoring: S n (1 r) = a 1 (1 r n ) And finally, S n n ( r ) a 1 1 = 1 r July 013 Page 8 of 64

29 c. Let s use this formula to find the total area of only the new additions through the 5 th stage. (What is a 1 in this case? What is n?) Check your answer by summing the values in your table. d. Now, add in the area of the original triangle. What is the total area of the Koch snowflake at the fifth stage? Do you think it s possible to find the area of the snowflake for a value of n equal to infinity? This is equivalent to finding the sum of an infinite geometric series. You ve already learned that we cannot find the sum of an infinite arithmetic series, but what about a geometric one? e. Let s look at an easier series: 1 + ½ + ¼ + Make a table of the first 10 sums of this series. What do you notice? Terms Sum 1 3/ 7/4 f. Now, let s look at a similar series: Again, make a table. How is this table similar or different from the one above? Why do think this is so? Terms Sum Recall that any real number -1 < r < 1 gets smaller when it is raised to a positive power; whereas numbers less than -1 and greater than 1, i.e. r > 1, get larger when they are raised to a positive power. n ( r ) a1 Thinking back to our sum formula, S 1 n =, this means that if r < 1, as n gets larger, r n 1 r approaches 0. If we want the sum of an infinite geometric series, we now have a1 ( 1 0) a1 S = =. We say that if a sum of an infinite series exists in this case, the sum of 1 r 1 r an infinite geometric series only exists if r < 1--- then the series converges to a sum. If an infinite series does not have a sum, we say that it diverges. g. Of the series in parts e and f, which would have an infinite sum? Explain. Find, using the formula above, the sum of the infinite geometric series. July 013 Page 9 of 64

30 h. Write out the formula for the sum of the first n terms of the sequence you summed in part g. Graph the corresponding function. What do you notice about the graph and the sum you found? i. Graphs and infinite series. Write each of the following series using sigma notation. Then find the sum of the first 0 terms of the series; write out the formula. Finally, graph the function corresponding to the sum formula for the first nth terms. What do you notice about the numbers in the series, the function, the sum, and the graph? ( ) ( ) ( ) j. Let s return to the area of the Koch Snowflake. If we continued the process of creating new triangles infinitely, could we find the area of the entire snowflake? Explain. k. If it is possible, find the total area of the snowflake if the iterations were carried out an infinite number of times. This problem is quite interesting: We have a finite area but an infinite perimeter! July 013 Page 30 of 64

31 Part Two: The Sierpinski Triangle (Images taken from Wikimedia Commons at Another example of a fractal is the Sierpinski triangle. Start with a triangle of side length 1. This time, we will consider the original picture as Stage 0. In Stage 1, divide the triangle into 4 congruent triangles by connecting the midpoints of the sides, and remove the center triangle. In Stage, repeat Stage 1 with the three remaining triangles, removing the centers in each case. This process repeats at each stage. 1. Mathematical Questions: Make a list of questions you have about this fractal, the Sierpinski triangle. What types of things might you want to investigate?. Number of Triangles in the Stages of the Sierpinski Fractal a. How many shaded triangles are there at each stage? How many removed triangles are there? Use the table to help organize your answers. Stage Number of Shaded Triangles Number of Newly Removed Triangles b. Are the sequences above the number of shaded triangles and the number of newly removed triangles arithmetic or geometric sequences? How do you know? c. How many shaded triangles would there be in the nth stage? Write both the recursive and explicit formulas for the number of shaded triangles at the nth stage. d. How many newly removed triangles would there be in the nth stage? Write both the recursive and explicit formulas for the number of newly removed triangles at the nth stage. July 013 Page 31 of 64

32 e. We can also find out how many removed triangles are in each evolution of the fractal. Write an expression for the total number of removed triangles at the nth stage. Try a few examples to make sure that your expression is correct. f. Find the total number of removed triangles at the 10 th stage. g. If we were to continue iterating the Sierpinski triangle infinitely, could we find the total number of removed triangles? Why or why not. If it is possible, find the sum. 3. Perimeters of the Triangles in the Sierpinski Fractal a. Assume that the sides in the original triangle are one unit long. Find the perimeters of the shaded triangles. Complete the table below. Stage Length of a Side of a Shaded Triangle Perimeter of each Shaded Triangle Number of Shaded Triangles n a. Find the perimeter of the shaded triangles in the 10 th stage. Total Perimeter of the Shaded Triangles b. Is the sequence of values for the total perimeter arithmetic, geometric, or neither? Explain how you know. c. Write recursive and explicit formulas for this sequence. In both forms, the common ratio should be clear. July 013 Page 3 of 64

33 4. Areas in the Sierpinski Fractal Assume that the length of the side of the original triangle is 1. Determine the exact area of each shaded triangle at each stage. Use this to determine the total area of the shaded triangles at each stage. (Hint: How are the shaded triangles at stage alike or different?) Stage Length of a Side of a Shaded Triangle (in) Area of each Shaded Triangle (in ) Number of Shaded Triangles n Total Area of the Shaded Triangles (in ) a. Explain why both the sequence of the area of each shaded triangle and the sequence of the total area of the shaded triangles are geometric sequences. What is the common ratio in each? Explain why the common ratio makes sense in each case. b. Write the recursive and explicit formulas for the sequence of the area of each shaded triangle. Make sure that the common ratio is clear in each form. c. Write the recursive and explicit formulas for the sequence of the total area of the shaded triangles at each stage. Make sure that the common ratio is clear in each form. d. Propose one way to find the sum of the areas of the removed triangles using the results above. Find the sum of the areas of the removed triangles in stage 5. July 013 Page 33 of 64

34 Stage e. Another way to find the sum of the areas of the removed triangles is to find the areas of the newly removed triangles at each stage and sum them. Use the following table to help you organize your work. Length of a Side of a Removed Triangle (in) Area of each Newly Removed Triangle (in ) Number of Newly Removed Triangles Total Area of the Newly Removed Triangles (in ) n f. Write the explicit and recursive formulas for the area of the removed triangles at stage n. g. Write an expression using summation notation for the sum of the areas of the removed triangles at each stage. Then use this formula to find the sum of the areas of the removed triangles in stage 5. h. Find the sum of the areas of the removed triangles at stage 0. What does this tell you about the area of the shaded triangles at stage 0? (Hint: What is the area of the original triangle?) i. If we were to continue iterating the fractal, would the sum of the areas of the removed triangles converge or diverge? How do you know? If it converges, to what value does it converge? Explain in at least two ways. July 013 Page 34 of 64

35 Part Three: More with Geometric Sequences and Series Up to this point, we have only investigated geometric sequences and series with a positive common ratio. We will look at some additional sequences and series to better understand how the common ratio impacts the terms of the sequence and the sum. 1. For each of the following sequences, determine if the sequence is arithmetic, geometric, or neither. If arithmetic, determine the common difference d. If geometric, determine the common ratio r. If neither, explain why not. a., 4, 6, 8, 10, d., 4, 8, 16, 3, b., -4, 6, -8, 10, e., -4, 8, -16, 3, c. -, -4, -6, -8, -10, f. -, -4, -8, -16, -3,. Can the signs of the terms of an arithmetic or geometric sequence alternate between positive and negative? Explain. i i 3. Write out the first 6 terms of the series ( ), ( ) notice? n i= 1 n n i, and ( ) 1 i 1 ( ) i= 0 i= 1. What do you 4. Write each series in summation notation and find sum of first 10 terms. a. 1 ½ + ¼ - 1/8 + b. 3 + ¾ + 3/16 + 3/64 + c Which of the series above would converge? Which would diverge? How do you know? For the series that will converge, find the sum of the infinite series. July 013 Page 35 of 64

36 Divide and Conquer Introduction In this task, we are going to perform division with polynomials and determine whether this operation is closed. We will make the connection of division with integers to division with polynomials by looking at the process of long division. We will then explore an alternate method for finding quotients when the divisor can be written in ( x k) form. Finally, we will explore an application of the remainder theorem by performing synthetic substitution in order to evaluate polynomials for given variables of interest. Materials Pencil Handout Graphing Calculator You may have noticed that in the first task, We ve Got to Operate, we only performed addition, subtraction, and multiplication of polynomials. Now we are ready to explore polynomial division. 1. First, let s think about something we learned in elementary school, long division. Find the quotient using long division and describe what you do in each step: a b July 013 Page 36 of 64

37 . Now, we are going to use the same idea to explore polynomial division. Specifically, read the example below and determine if this process is similar to the methods you described in your process of long division above. Polynomial Long Division If you're dividing a polynomial by something more complicated than just a simple monomial, then you'll need to use a different method for the simplification. That method is called "long (polynomial) division", and it works just like the long (numerical) division you did back in elementary school, except that now you're dividing with variables. Divide x 9x 10 by x + 1 Think back to when you were doing long division with plain old numbers. You would be given one number that you had to divide into another number. You set up the division symbol, inserted the two numbers where they belonged, and then started making guesses. And you didn't guess the whole answer right away; instead, you started working on the "front" part (the larger place values) of the number you were dividing Long division for polynomials works in much the same way: \ First, I set up the division: For the moment, I'll ignore the other terms and look just at the leading x of the divisor and the leading x of the dividend. If I divide the leading x inside by the leading x in front, what would I get? I'd get an x. So I'll put an x on top: Now I'll take that x, and multiply it through the divisor, x + 1. First, I multiply the x (on top) by the x (on the "side"), and carry the x underneath: Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath: July 013 Page 37 of 64

38 Then I'll draw the "equals" bar, so I can do the subtraction. To subtract the polynomials, I change all the signs in the second line......and then I add down. The first term (the x ) will cancel out: I need to remember to carry down that last term, the "subtract ten", from the dividend: Now I look at the x from the divisor and the new leading term, the 10x, in the bottom line of the division. If I divide the 10x by the x, I would end up with a 10, so I'll put that on top: Now I'll multiply the 10 (on top) by the leading x (on the "side"), and carry the 10x to the bottom:...and I'll multiply the 10 (on top) by the 1 (on the "side"), and carry the 10 to the bottom: I draw the equals bar, and change the signs on all the terms in the bottom row: July 013 Page 38 of 64

39 Then I add down: Then the solution to this division is: x 10 Since the remainder on this division was zero (that is, since there wasn't anything left over), the division came out "even". When you do regular division with numbers and the division comes out even, it means that the number you divided by is a factor of the number you're dividing. For instance, if you divide 50 by 10, the answer will be a nice neat "5" with a zero remainder, because 10 is a factor of 50. In the case of the above polynomial division, the zero remainder tells us that x + 1 is a factor of x 9x 10, which you can confirm by factoring the original quadratic dividend, x 9x Your teacher will now guide you through several of these to practice long division. 3 a. x + 3x 1 x + 5x + 5x b. x + 3 6x + x c. ( x + 6x 5x + 0) ( x + 5) d. (4x 3x 7x 4x 9) ( x ) 4. As you can see, long division can be quite tedious. Now, let us consider another way to find quotients called synthetic division. Unfortunately, synthetic division is defined only when the divisor is linear. a. In which problem(s) above is synthetic division defined? b. The next part of this task will explore how it works and why it only works when there is a linear divisor. July 013 Page 39 of 64

40 The following excerpt is taken from: J.M. Kingston, Mathematics for Teachers of the Middle Grades, John Wiley & Sons, Inc., NY, 1966, p July 013 Page 40 of 64

41 July 013 Page 41 of 64

42 5. Your teacher will now guide you through several of these to practice synthetic division. 3 3 a. x x + x 6x 9 b. x + 3 x + x 6x c. ( x 16x + x + 4) ( x + 4) d. (4x 3x 7x 4x 9) ( x ) 6. Compare you answers in problem 3d to your solution to 5d. Is it the same? Why? 7. Look back at all of your quotients in problems 3 and 5. Is the operation of division closed for polynomials? In other words, are the results of the division operation always an element in the set of polynomials? Why or why not? 8. One way to evaluate polynomial functions is to use direct substitution. For instance, 4 f ( x) = x 8x + 5x 7 can be evaluated when x = 3 as follows: 4 f (3) = (3) 8(3) + 5(3) 7 = = 98. However, there is another way to evaluate a polynomial function called synthetic substitution. Since the Remainder Theorem states that the remainder of a polynomial f(x) divided by a linear divisor (x c) is equal to f(c), the value of the last number on the right corner should give an equivalent result. Let s see We can see that the remainder is equivalent to the solution to this problem! Use synthetic substitution to evaluate the following. You can confirm your results with direct substitution using a calculator. July 013 Page 4 of 64

43 4 3 a. f ( x) = x + x 3x + 5x 8, x = 1 3 b. f ( x) = 3x + 7x 4x + 8, x = 3 5 c. f ( x) = 3x x + x, x = 4 3 d. f ( x) = x + 8x + 13x 4, x = July 013 Page 43 of 64

44 Factors, Zeros, and Roots: Oh My! Solving polynomials that have a degree greater than those solved in Coordinate Algebra and Analytic Geometry is going to require the use of skills that were developed when we solved quadratics last year. Let s begin by taking a look at some second degree polynomials and the strategies used to solve them. These equations have the form ax + bx + c = 0, and when they are graphed the result is a parabola. 1. Factoring is used to solve quadratics of the form ax + bx + c = 0 when the roots are rational. Find the roots of the following quadratic functions: a. f ( x) = x 5x 14 b. f ( x) = x 64 c. f ( x) = 6x + 7x 3 d. f ( x) = 3x + x. Another option for solving a quadratic whether it is factorable but particularly when it is not is to use the quadratic formula. Remember, a quadratic equation written in ax + bx + c = 0 has b ± b 4ac solution(s) x = a Also remember that b 4ac is the discriminant and gives us the ability to determine the nature of the roots. > 0 real roots b 4ac = 0 1 real root < 0 0 real roots ( imaginary) Find the roots for each of the following. Also, describe the number and nature of these roots. a. f ( x) = 4x x + 9 July 013 Page 44 of 64

45 b. f ( x) = 3x + 4x 8 c. f ( x) = x 5x Let s take a look at the situation of a polynomial that is one degree greater. When the polynomial is a third degree, will there be any similarities when we solve? 3 Suppose we want to find the roots of f ( x) = x + x 5x 6. By inspecting the graph of the function, we can see that one of the roots is distinctively. Since we know that x = is a solution to f (x), we also know that x is a factor of the expression x 3 + x 5x 6. This means that if we divide x 3 + x 5x 6 by x there will be a remainder of zero. Let s confirm this with synthetic substitution: Let s practice synthetic division before we tackle how to solve cubic polynomials in general. Do the following division problems synthetically. a. 10x 3 17x 7x + x b. x + x 10x 4 x c. 3 x 7x 6 x + 1 July 013 Page 45 of 64

46 The main thing to notice about solving cubic polynomials (or higher degree polynomials) is that a polynomial that is divisible by x k has a root at k. Synthetic division applied to a polynomial and a factor result in a zero for the remainder. This leads us to the Factor Theorem, which states: A polynomial f (x) has a factor x k if and only if f ( k) = 0. Solving cubic polynomials can be tricky business sometimes. A graphing utility can be a helpful tool to identify some roots, but in general there is no simple formula for solving cubic polynomials like the quadratic formula aids us in solving quadratics. There is however a tool that we can use for helping us to identify Rational Roots of the polynomial in question. 4. The Rational Root Theorem states that any rational solutions to a polynomial will be in the form of q p where p is a factor of the constant term of the polynomial (the term that does not show a variable) and q is a factor of the leading coefficient. This is actually much simpler than it appears at first glance. 3 a. Let us consider the polynomial f ( x) = x 5x 4x + 0 Identify p (all the factors of the constant term 0) = Identify q (all the factors of the leading coefficient 1) = Identify all possible combinations of q p : 3 If f ( x) = x 5x 4x + 0 is going to factor, then one of these combinations is going to work, that is, the polynomial will divide evenly. So the best thing to do is employ a little trial and error. Let s start with the smaller numbers, they will be easier to evaluate. Substitute for x: 1, -1,, -, 4, -4 0, -0. Why would substituting these values in for x be a useful strategy? Why do we not have to use synthetic division on every one? July 013 Page 46 of 64

47 Define what the Remainder Theorem states and how it helps us. Hopefully, you did not get all the way to -0 before you found one that works. Actually, should have worked. Once there is one value that works, we can go from there. Use the factor ( x ) to divide f (x). This should yield: 3 f ( x) = x 5x 4x + 0 = ( x )( x 3x 10) 3 By factoring the result we can find all the factors: f ( x) = x 5x 4x + 0 = ( x )( x + )( x 5) Therefore the roots are, -, and 5. What could be done if this portion was not factorable? 5. Use the Quadratic Formula For each of the following find each of the roots, classify them and show the factors. 3 a. f ( x) = x 5x 4x + 0 Possible rational roots: Show work for Synthetic Division and Quadratic Formula (or Factoring): Complete Factorization: Roots and Classification Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary July 013 Page 47 of 64

48 3 b. f ( x) = x + x 5x 6 Possible rational roots: Show work for Synthetic Division and Quadratic Formula(or Factoring): Complete Factorization: Roots and Classification Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary 3 c. f ( x) = 4x 7x + 3 Possible rational roots: Show work for Synthetic Division and Quadratic Formula (or Factoring): Complete Factorization: Roots and Classification Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary July 013 Page 48 of 64

49 6. What happens when we come to a function that is a 4 th degree? Well, just like the cubic there is no formula to do the job for us, but by extending our strategies that we used on the third degree polynomials, we can tackle any quartic function. 1 st Develop your possible roots using the q p method. nd Use synthetic division with your possible roots to find an actual root. If you started with a 4 th degree, that makes the dividend a cubic polynomial. 3 rd Continue the synthetic division trial process with the resulting cubic. Don t forget that roots can be used more than once. 4 th Once you get to a quadratic, use factoring techniques or the quadratic formula to get to the other two roots. For each of the following find each of the roots, classify them and show the factors. 4 3 a. f ( x) = x + x 9x x + 8 Possible rational roots: Show work for Synthetic Division and Quadratic Formula (or Factoring): Complete Factorization: Roots and Classification Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary 4 3 b. f ( x) = x 11x 13x + 11x + 1 Possible rational roots: Show work for Synthetic Division and Quadratic Formula(or Factoring): July 013 Page 49 of 64

50 Complete Factorization: Roots and Classification Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary c. f ( x) = x 1x + 49x 90x + 76x 4 Possible rational roots: Show work for Synthetic Division and Quadratic Formula(or Factoring): Complete Factorization: Roots and Classification Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary July 013 Page 50 of 64

51 5 4 3 d. f ( x) = x 5x + 8x 8x + 16x 16 Possible rational roots: Show work for Synthetic Division and Quadratic Formula (or Factoring): Complete Factorization: Roots and Classification Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary 7. Let s consider a scenario where the roots are imaginary. 4 3 Suppose that you were asked to find the roots of f ( x) = x x + 3x 4x 4. There are only 6 possible roots: ± 1, ±, ± 4. In the light of this fact, let s take a look at the graph of this function. It should be apparent that none of these possible solutions are roots of the function. And without a little help at this point we are absolutely stuck. None of the strategies we have discussed so far help us at this point. a. But consider that we are given that one of the roots of the function is i. Because roots come in pairs (think for a minute about the quadratic formula); an additional root should be -i. So, let s take these values and use them for synthetic division. July 013 Page 51 of 64

52 b. Though the values may not be very clean, this process should work just as it did earlier. Take a moment and apply what you have been doing to this function. Complete Factorization: Roots and Classification Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary Rational Irrational Real Imaginary July 013 Page 5 of 64

53 Polynomials Patterns Task 1. To get an idea of what polynomial functions look like, we can graph the first through fifth degree polynomials with leading coefficients of 1. For each polynomial function, make a table of 6 points and then plot them so that you can determine the shape of the graph. Choose points that are both positive and negative so that you can get a good idea of the shape of the graph. Also, include the x intercept as one of your points. a. For example, for the first order polynomial function: y = x 1. You may have the following table and graph: x y b. y = x c. y = x 3 x y x y July 013 Page 53 of 64

54 d. y = x 4 e. y = x 5 x y x y f. Compare these five graphs. By looking at the graphs, describe in your own words how y = x is different from y = x 4. Also, how is y = x 3 different from y = x 5? g. Note any other observations you make when you compare these graphs.. In order to examine their characteristics in detail so that we can find the patterns that arise in the behavior of polynomial functions, we can study some examples of polynomial functions and their graphs. Here are 8 polynomial functions and their accompanying graphs that we will use to refer back to throughout the task. July 013 Page 54 of 64

55 Handout of Graphs of Polynomial Functions : f(x) = x + x; f(x)= x(x+) k(x) = x 4 5x +4 ; k(x) = (x-1)(x+1)(x-)(x+) g(x) = x + x; g(x)=x(-x+1) l(x) = (x 4 5x +4) ; l(x) = -(x-1)(x+1)(x-)(x+) h(x) = x 3 x ; h(x) = x(x 1)(x + 1) m(x) = 1 (x5 + 4x 4 7x 3 x + 4x); m(x)= 1 x(x-1)(x-)(x+3)(x+4) j(x)= x 3 + x + 3x; j(x) = -x(x-3)(x+1) n(x) = 1 (x5 + 4x 4 6x 3 15x + 36x); n(x)=- 1 x(x-1)(x-)(x+3)(x+4) July 013 Page 55 of 64

56 Each of these equations can be re-expressed as a product of linear factors by factoring the equations, as shown below in the gray equations. a. List the x-intercepts of j(x) using the graph above. How are these intercepts related to the linear factors in gray? b. Why might it be useful to know the linear factors of a polynomial function? c. Although we will not factor higher order polynomial functions in this unit, you have factored quadratic functions in Coordinate Geometry. For review, factor the following second degree polynomials, or quadratics. y = x x 1 y = x + 5x 6 y = x 6x 10 d. Using these factors, find the roots of these three equations. e. Sketch a graph of the three quadratic equations above without using your calculator and then use your calculator to check your graphs. f. You can factor some polynomial equations and find their roots in a similar way. Try this one:. y = x 5 + x 4 x 3. What are the roots of this fifth order polynomial function? g. How many roots are there? Why are there not five roots since this is a fifth degree polynomial? July 013 Page 56 of 64

57 h. Check the roots by generating a graph of this equation using your calculator. i. For other polynomial functions, we will not be able to draw upon our knowledge of factoring quadratic functions to find zeroes. For example, you may not be able to factor = x 3 + 8x + 5x 14, but can you still find its zeros by graphing it in your calculator? How? Write are the zeros of this polynomial function. 3. Symmetry The first characteristic of these 8 polynomials functions we will consider is symmetry. a. Sketch a function you have seen before that has symmetry about the y-axis. Describe in your own words what it means to have symmetry about the y-axis. What is do we call a function that has symmetry about the y-axis? b. Sketch a function you have seen before that has symmetry about the origin. Describe in your own words what it means to have symmetry about the origin. What do we call a function that has symmetry about the origin? July 013 Page 57 of 64

58 c. Using the table below and your handout of the following eight polynomial functions, classify the functions by their symmetry. Function f(x) = x + x g(x) = x + x h(x) = x 3 x j(x)= x 3 + x + 3x k(x) = x 4 5x +4 Symmetry about the y axis? Symmetry about the origin? Even, Odd, or Neither? l(x) = (x 4 5x +4) m(x) = 1 (x5 + 4x 4 7x 3 x + 4x) n(x) = 1 (x5 + 4x 4 7x 3 x + 4x) d. Now, sketch your own higher order polynomial function (an equation is not needed) with symmetry about the y-axis. e. Now, sketch your own higher order polynomial function with symmetry about the origin. f. Using these examples from the handout and the graphs of the first through fifth degree polynomials you made, why do you think an odd function may be called an odd function? Why are even functions called even functions? g. Why don t we talk about functions that have symmetry about the x-axis? Sketch a graph that has symmetry about the x-axis. What do you notice? July 013 Page 58 of 64

59 4. Domain and Range Another characteristic of functions that you have studied is domain and range. For each polynomial function, determine the domain and range. Function Domain Range f(x) = x + 3 g(x) = x h(x) = x 3 x j(x)= x 3 + x + 3x k(x) = x 4 5x +4 l(x) = (x 4 5x +4) m(x) = 1 (x5 + 4x 4 7x 3 x + 4x) n(x) = 1 (x5 + 4x 4 7x 3 x + 4x) 5. Zeros a. We can also describe the functions by determining some points on the functions. We can find the x-intercepts for each function as we discussed before. Under the column labeled xintercepts write the ordered pairs (x,y) of each intercept and record the number of intercepts in the next column. Also record the degree of the polynomial. Function Degree X-intercepts Zeros Number of Zeros f(x) = x + x g(x) = x + x h(x) = x 3 x j(x)= x 3 + x + 3x k(x) = x 4 5x +4 l(x) = (x 4 5x +4) m(x) = 1 (x5 + 4x 4 7x 3 x + 4x) n(x) = 1 (x5 + 4x 4 7x 3 x + 4x) b. These x-intercepts are called the zeros of the polynomial functions. Why do you think they have this name? c. Fill in the column labeled Zeroes by writing the zeroes that correspond to the x-intercepts of each polynomial function, and also record the number of zeroes each function has. July 013 Page 59 of 64

60 d. Make a conjecture about the relationship of degree of the polynomial and number of zeroes. e. Test your conjecture by graphing the following polynomial functions using your calculator: y = x, y = x (x 1)(x + 4), y = x(x 1). Function Degree X-Intercepts Zeroes Number of Zeroes y = x (0,0) y = x (x 1)(x + 4) (0,0); (0,-1);(0-4) y = x(x 1) How are these functions different from the functions in the table? Now amend your conjecture about the relationship of the degree of the polynomial and the number of x-intercepts. Make a conjecture for the maximum number of x-intercepts the following polynomial function will have: p(x) = x x 6 3x 6. End Behavior In determining the range of the polynomial functions, you had to consider the end behavior of the functions, that is the value of f (x) as x approaches infinity and negative infinity. Polynomials exhibit patterns of end behavior that are helpful in sketching polynomial functions. a. Graph the following on your calculator. Make a rough sketch next to each one and answer the following: Is the degree even or odd? Is the leading coefficient, the coefficient on the term of highest degree, positive or negative? Does the graph rise or fall on the left? On the right? y = x. y = x y = 3x 9. y = x y = 3x y = x 3 4 July 013 Page 60 of 64

61 4 4. y = 5x y = x y = x 1. y = x y = 3x y = 7x b. Write a conjecture about the end behavior, whether it rises or falls at the ends, of a function of the form f(x) = ax n for each pair of conditions below. Then test your conjectures on some of the 8 polynomial functions graphed on your handout. Condition a: When n is even and a > 0, Condition b: When n is even and a < 0, Condition c: When n is odd and a > 0, Condition d: When n is odd and a < 0, c. Based on your conjectures in part (b), sketch a fourth degree polynomial function with a negative leading coefficient. d. Now sketch a fifth degree polynomial with a positive leading coefficient. July 013 Page 61 of 64

62 Note we can sketch the graph with the end behavior even though we cannot determine where and how the graph behaves otherwise without an equation or without the zeros. e. If we are given the real zeros of a polynomial function, we can combine what we know about end behavior to make a rough sketch of the function. Sketch the graph of the following functions using what you know about end behavior and zeros: f. f(x) = (x )(x 3) g. f(x) = x(x 1)(x + 5)(x 7) 7. Critical Points Other points of interest in sketching the graph of a polynomial function are the points where the graph begins or ends increasing or decreasing. Recall what it means for a point of a function to be an absolute minimum or an absolute maximum. a. Which of the twelve graphs from part 6a have an absolute maximum? b. Which have an absolute minimum? c. What do you notice about the degree of these functions? d. Can you ever have an absolute maximum AND an absolute minimum in the same function? If so, sketch a graph with both. If not, why not? July 013 Page 6 of 64

63 e. For each of the following graphs from the handout, locate the turning points and the related intervals of increase and decrease, as you have determined previously for linear and quadratic polynomial functions. Then record which turning points are relative minimum (the lowest point on a given portion of the graph) and relative maximum (the highest point on a given portion of the graph) values. Function Degree Turning Points f(x) h(x) k(x) n(x) Intervals of Increase Intervals of Decrease Relative Minimum Relative Maximum f. Make a conjecture about the relationship of the degree of the polynomial and the number of turning points that the polynomial has. Recall that this is the maximum number of turning points a polynomial of this degree can have because these graphs are examples in which all zeros have a multiplicity of one. g. Sometimes points that are relative minimums or maximums are also absolute minimums or absolute maximum. Are any of the relative extrema in your table also absolute extrema? 8. Putting it all Together: Now that you have explored various characteristics of polynomial functions, you will be able to describe and sketch graphs of polynomial functions when you are given their equations. a. If I give you the function: f ( x) = ( x 3)( x 1) then what can you tell me about the graph of this function? Make a sketch of the graph of this function, describe its end behavior, and locate its critical point and zeroes. July 013 Page 63 of 64

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