Polynomials Classwork

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1 Polynomials Classwork

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3 What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable is a function defined by are the The of a polynomial is the of the of. Def.: The of a quadratic (polynomial) equation is the value of when. ** We also call the of the of. ** Ex.: Find the zeros of. 1

4 Three Observations about Polynomials with degree 1. It is a smooth curve. 2. It is continuous. 3. The leading term dominates when is large. The graph rises or falls without bounds as Ex.: What is the end behavior of each polynomial? (a) (b) Ex.: Sketch the graph of. 2

5 Zeros of Polynomial Functions Numerical, Analytical and Graphical Approaches For each of the functions graphed below, state the end behavior and the zeros of the functions. Graph of f(x) Graph of g(x) Graph of h(x) f(x) is a function. g(x) is a function. h(x) is a function. Left End Behavior Left End Behavior Left End Behavior Right End Behavior Right End Behavior Right End Behavior Zeros Zeros Zeros Define the multiplicity of a zero. Read the following information about the multiplicities of the zeros of f(x), g(x), and h(x) while studying the graphs above. Then, answer the questions on the next page. In the graph of f(x), all of the zeros have a multiplicity of 1. In the graph of g(x), the zero of x = 2 has a multiplicity of 1 and x = 2 has a multiplicity of 3. In the graph of h(x), the zeros x = 4, x = 2, and x = 5 have a multiplicity of 1 and x = 2 has a multiplicity of 2. 3

6 1. What do you notice about the sum of the multiplicities of the zeros and the degree of the function? 2. Describe the behavior of the graph as it approaches a zero whose multiplicity is Describe the behavior of the graph as it approaches a zero whose multiplicity is Describe the behavior of the graph as it approaches a zero whose multiplicity is 3. Fundamental Theorem of Algebra: Every polynomial of degree zero (root), possibly imaginary. n 0has at least one Corollary: A polynomial function of degree n has EXACTLY ZEROS. Theorem: Every polynomial of degree n 0 can be written as a product of a constant and n linear factors. 4

7 Examples: (a) f(x) = x 3 + 2x 2 x 2 Type of function: Root: Multiplicity: Describe the behavior of the graph at this root. Root: Multiplicity: Describe the behavior of the graph at this root. Root: Multiplicity: Describe the behavior of the graph at this root. (b) h(x) = 2x 3 x 2 4x + 3 Type of function: Root: Multiplicity: Describe the behavior of the graph at this root. Root: Multiplicity: Describe the behavior of the graph at this root. (c) p(x) = 2x 4 7x 3 6x x 40 Type of function: Root: Multiplicity: Describe the behavior of the graph at this root. Root: Multiplicity: Describe the behavior of the graph at this root. 5

8 Now, let s consider how we might be able to locate the zeros of a polynomial function numerically. Consider the function h(x) = 2x 3 x 2 4x + 3 that we investigated earlier and whose graph is shown below. Find each pair of function values in the table below and answer the questions that follow Find h( 2) and h( 1). Find h(0) and h(2). From the graph, clearly h(x) has a zero between x = 2 and x = 1. Explain how your finding the values of h( 2) and h( 1) above numerically shows that there is a zero that exists between x = 2 and x = 1. Does the same reasoning that you described concerning the zero between x = 2 and x = 1 hold true for the existence of a zero between x = 0 and x = 2? Explain your reasoning. Based on what we have just seen, what inference can you make about the existence of a zero of a polynomial function if you know the value of the function at two different x values? 6

9 Ex.: The table of values below represents values of a cubic function. The function has a negative zero and two positive zeros. Answer the questions that follow. x F(x) (a) Is/Are any of the zeros of F(x) specifically identified in the table? Explain your reasoning. (b) Between which two x values in the table is the negative zero located? Explain your reasoning. (c) Between which two x values in the table is the second positive zero located? Explain your reasoning. 7

10 The Remainder Theorem of Polynomial Functions I. Divide 12,372 by 11. Then, identify the divisor, the dividend, the quotient, and the remainder. Divisor: Dividend: Quotient: Remainder: Answer to the division problem: II. (a) Divide by. Method 1: Long Division Method 2: (b) Divisor: Dividend: Quotient: Remainder: Answer to the division problem: (c) Division Algorithm: 8

11 5x 4 7x x x 1 2 4x x 1 x 5 3x 2 x 4 2x x 13x 6 2x 1 Perform synthetic division Perform synthetic division Perform synthetic division Divisor Divisor Divisor Dividend Dividend Dividend Quotient Quotient Quotient Remainder Remainder Remainder Answer to the division Answer to the division Answer to the division From the five previous examples you should realize that there is a relationship between the remainder when a polynomial, P(x), is divided by a linear binomial, (x a), and the value of the function P(a). What is the relationship? 9

12 Complete the following statement. Remainder Theorem If a polynomial function, f(x), is divided by a factor, (x r), then the remainder will be the same value as. 1. Find the remainder when is divided by. 2. Find the remainder when is divided by. 3. For what value of k will the function P(x) = 2x 3 2x 2 + kx 2 have a remainder of 8 when divided by the factor (x + 2)? 4. For what value of k will the function P(x) = 3x 3 + kx 2 5 have a remainder of 4 when divided by the factor (x 3)? 5. For what value of k will the function P(x) = x 4 2x 2 + kx 6 have a remainder of 0 when divided by the factor (x + 1)? 6. Use synthetic division to find and if. 10

13 Factor Thm, Fundamental Thm of Algebra, and End Behavior I. Factor Theorem: The complex # c is a zero of a polynomial p ( c) 0 p(x). p(x) if and only if x c is a factor of 1. Is 3 p ( x) 2x 5x? x 2 a factor of 6 2. Determine which of the following factors are factors of the function, Show and explain your work. A. B. C. 3. The function in number 2 is a cubic function whose highest exponent is 3. Thus, there should be 3 zeros of the function. Based on your work in number 2, what is the other zero of? What is the multiplicity of each root? Explain your reasoning. 4. Find the values of k for which x is a factor of p ( x) 3x 2x 10x 3kx

14 II. For each of the functions in the chart below, graph the polynomial function, paying close attention as to how the function comes into the view screen and exits the view screen of the calculator. Based on your observations of the function being graphed, fill in the chart below. Function Even or Odd Degree Positive or Negative Leading Coefficient Rise or Fall to the Left Rise or Fall to the Right 1. f(x) = 2x 2 3x f(x) = x 4 + x 3 x f(x) = x 2 3x f(x) = x 4 x 3 + 2x f(x) = x 3 3x 2 + 2x 1 6. f(x) = x 5 x 3 + 2x f(x) = x 3 + 2x 2 x f(x) = x 5 + 4x 4 3x 3 + x f(x) = 2x 4 x 3 + 8x x Now, write some conjectures about the end behavior of polynomial functions. Even degree Left & Right End Behavior is. Odd degree Left & Right End Behavior is. 1. When and the degree is even, the graph. 2. When and the degree is odd, the graph. 3. When and the degree is even, the graph. 4. When and the degree is odd, the graph. 12

15 13

16 The fact of the matter is that you already knew this information based on what you know about the graphical behavior of linear and quadratic functions. For a linear function f(x) = ax + b, the degree of the function is odd. Sketch a linear function in which a < 0 and one in which a > 0. Then, identify the end behavior. For a quadratic function f(x) = ax 2 + bx + c, the degree of the function is even. Sketch a quadratic function in which a < 0 and one in which a > 0. Then, identify the end behavior. V. For each of the graphed functions below, identify the zeros, the degree, and determine if the leading coefficient of the equation of the function is positive or negative. Justify your reasoning for the sign of the leading coefficient. Graph of the Function Zeros and Their Multiplicities Degree and Type of Function Left and Right End Behavior Positive or Negative Leading Coefficient 14

17 Graph of the Function Zeros and Their Multiplicities Degree and Type of Function Left and Right End Behavior Positive or Negative Leading Coefficient 15

18 Ex. 1: The table below shows function values of a polynomial function. The zeros in the table are the only zeros of the function, and no zeros have a multiplicity greater than two. Answer the questions that follow. x F(x) (a) Identify the left and right end behavior of the function, F(x). (b) Based on the end behavior, what must be true about the degree of the function? Give a reason for your answer. (c) Identify the negative zero of the function. What is the multiplicity of this zero? Give a reason for your answer. (d) Identify the positive zero of the function. What is the multiplicity of this zero? Give a reason for your answer. 16

19 Ex. 2: The table below shows function values of a polynomial function. The zeros in the table are the only zeros of the function, and no zeros have a multiplicity greater than two. Answer the questions that follow. x G(x) (a) Identify the left and right end behavior of the function, F(x). (b) Based on the end behavior, what must be true about the degree of the function? Give a reason for your answer. (c) Identify the negative zeros of the function. What is the multiplicity of these zeros? Give a reason for your answer. (d) Identify the positive zero of the function. What is the multiplicity of this zero? Give a reason for your answer. (e) What type of polynomial is? 17

20 Ex. 3: Sketch a possible graph of each of the following functions on the axes provided. Pay attention to the zeros of the function, making sure that the graph behaves appropriately at each based on their multiplicities and that the graph adheres to the appropriate end behavior. f(x) = (x + 2)(x 1)(x 3) g(x) = (x + 1)(x + 1)(x 2) h(x) = (x + 2)(x + 2)(x 2)(x 2) k(x) = (x + 3)(x 1)(x 1)(x 1) j(x) = (x + 2)(x + 2)(x 1)(x 1)(x 1) p(x) = (x + 3)(x + 1)(x + 1)(x 2)(x 2) 18

21 f(x) = (x + 2)(x 1) 2 (x 3) g(x) = (x + 1)(x + 1)(3 x) h(x) = (x + 3) 3 (x 2) k(x) = (x + 3)(3 2x)(x 3) 2 19

22 Analysis of Polynomial Functions Numerical, Graphical and Analytical Approaches x F(x) The table above shows function values for selected values in the domain of F(x), which is a quartic function. The leading coefficient F(x) is positive. (a) Identify the zeros of F(x). Give a numerical reason for your answer. (b) What is the multiplicity of each zero listed in part a. Give a reason for your answer. (c) Around what x value(s) does F(x) reach a relative minimum? Give a numerical reason for your answer. (d) Around what x value(s) does F(x) reach a relative maximum? Give a numerical reason for your answer. (e) Sketch a possible graph of F(x) 20

23 Intermediate Value Theorem: If a function is continuous on anf is a real number such that, then there exists at least one real number such that and. In other words, a function that is continuous on a closed interval takes on every value between and f(b). Ex.: Let. (a) Find: (b) Is there a root between and? Why? Location Principle: If is a continuous function and and have opposite signs for, then has at least one zero between and. 21

24 Using the PolyRoots Command to Approximate Radical Roots Let s find the roots for the polynomial. TI-Nspire: (Note: This method will give ALL of the roots at once.) Menu 3. Algebra 3. Polynomial Tools 3. Complex Roots of Polynomials Enter the polynomial, variable Press Enter. Solve: (Note: This method must be done for EACH root.) 2 nd Catalog Solve Form: Solve variable, guess, {lower, upper}) Ex.: Solve Bounds for Zeros: Upper Bound: all zeros are less than the upper bound. Least Upper Bound: Smallest number so that all zeros are less. Lower Bound: all zeros are greater than the lower bound. Greatest Lower Bound: Biggest number so that all zeros are greater. From previous example: Least Upper Bound: Greatest Lower Bound: 22

25 2. Pictured below are selected function values for a quartic polynomial function,, whose two zeros are real numbers. For all values on the intervals and, the function is such that (a) Are there any zeros of the function listed in the table? Justify your answer. (b) Between what two values does have other zeros? Justify your answer. (c) What can be concluded about one of the zeros of? Give a reason for your answer. (d) Can anything be concluded about the leading coefficient of? Justify your answer. (e) Sketch a possible graph of. 23

26 3. The equation of a cubic polynomial function is and. (a) Determine the left and right hand behavior of the graph of. Justify your answer. (b) If is divided by the factor, what will the remainder be? Justify your answer. (c) Rewrite the function in completely factored form. Show your work. (d) Identify the zeros of, stating the multiplicity of each root. Then, use the information to draw a possible sketch of. 24

27 Properties of Graphs of Polynomial Functions Define each of the following terms associated with polynomial functions. 1. Cubic Function 2. Quartic Function 3. Quintic Function 4. Intervals of increasing function values 5. Intervals of decreasing function values 6. Relative Maximum 7. Relative Minimum 8. Absolute Maximum 9. Absolute Minimum 10. Intervals of concave up 11. Intervals of concave down 12. Point of Inflection 25

28 Explain why the graph below is a cubic function. Explain why the graph below is a quartic function. Explain why the graph below is a quintic function Identify the intervals of where the graph below is increasing. Identify the intervals where the graph below is decreasing. Identify the coordinates of the relative maximum(s). 26

29 Identify the coordinates of the relative minimum(s). Identify the coordinates of the absolute maximum(s). Identify the coordinates of the absolute maximum(s). Identify the coordinates of the absolute minimum(s). Identify the coordinates of the absolute minimum(s). How many points of inflection does the graph have? Label them on the graph. 27

30 Identify the coordinates of the point(s) of inflection. Identify the intervals where the graph below is concave up and where it is concave down. Identify the coordinates of the point(s) of inflection. Identify the intervals where the graph below is concave up and where it is concave down. 28

31 Domain: Range: Intervals of x values where f(x) > 0: Intervals of x values where f(x) < 0: Left End Behavior: Right End Behavior: Zeros and their multiplicity: Type of Function: Possible Equation: Intervals of x values where f(x) is increasing: Intervals of x values where f(x) is decreasing: Relative Maximum(s): Relative Minimum(s): Absolute Maximum(s): Absolute Minimum(s): Point(s) of Inflection: Intervals of x values where f(x) is concave up: Intervals of x values where f(x) is concave down: 29

32 Writine An Equation When Given the Roots Ex.: Write a polynomial of lowest degree with roots and. 30

33 Domain: Range: Intervals of x values where f(x) > 0: Intervals of x values where f(x) < 0: Left End Behavior: Right End Behavior: Zeros and their multiplicity: Type of Function: Possible Equation: Intervals of x values where f(x) is increasing: Intervals of x values where f(x) is decreasing: Relative Maximum(s): Relative Minimum(s): Absolute Maximum(s): Absolute Minimum(s): Point(s) of Inflection: Intervals of x values where f(x) is concave up: Intervals of x values where f(x) is concave down: 31

34 Polynomial Functions that Have Imaginary Roots Descartes Rule of Signs The degree of a polynomial function identifies the number of roots that it will have. However, not all of the roots of the function have to be real roots. It is possible that some of the roots will be imaginary. Consider the two functions that follow to answer the given questions. If the graph is shifted up or down, what is the maximum number of zeros that the graph could have? Based on the maximum number of zeros that the graph could have, what type of function is graphed? f(x) Based on the graph, identify the number of positive, negative, zero and imaginary roots of the function. State the left and right end behavior of the function. Based on the degree and the end behavior, is the leading coefficient of the equation positive or negative? Based on the graph, what factor(s) is/are guaranteed to be factor(s) of the equation of f(x)? What are the domain and range of f(x)? 32

35 If the graph is shifted up or down, what is the maximum number of zeros that the graph could have? Based on the maximum number of zeros that the graph could have, what type of function is graphed? Based on the graph, identify the number of positive, negative, zero and imaginary roots of the function. State the left and right end behavior of the function. g(x) Based on the degree and the end behavior, is the leading coefficient of the equation positive or negative? Based on the graph, what factor(s) is/are guaranteed to be factor(s) of the equation of g(x)? What are the domain and range of g(x)? 33

36 Before we get to the star attraction of this lesson, let s do one more investigation. Graph each of the functions below on the graphing calculator. Draw a rough sketch of their graphs. 3 2 f ( x) x x 2x 3 2 g( x) x 3x h ( x) x x 4x Rewrite the function in completely factored form. Rewrite the function in completely factored form. Rewrite the function in completely factored form. Based on the factors, how many times is x = 0 a root of the function? How is this visible in the graph? Based on the factors, how many times is x = 0 a root of the function? How is this visible in the graph? Based on the factors, how many times is x = 0 a root of the function? How is this visible in the graph? Two of the three functions have something in common, both graphically and analytically, that is different from the other function. What is this difference? 34

37 Descartes Rule of Signs The # of positive real zeros of of changes in sign of the coefficients of or is less than this # by an even #. The # of negative real zeros of of changes in sign of the coefficients of or is less than this # by an even #. h ( x) 2x 8x 2x Consider the function given by the equation 3. Find the equation of x Descartes Rule of Signs is dependent upon the number of times the signs of the terms in the equation change signs. In the boxes below, write both the equation of h(x) and the equation of h( x). Then, count the number of times that the signs of the terms change. The number of sign changes in h(x) determine the MAXIMUM number of POSITIVE real roots that the function COULD have. The number of sign changes in h( x) determine the MAXIMUM number of NEGATIVE real roots that the function COULD have The function can have this maximum number of positive or negative roots or any multiple of two less than this maximum. For example, if a function can have a maximum of 4 positive roots, then it could also possible have 2 or 0 positive roots. If a function can have a maximum of 3 positive roots, then it could also have 1 positive root. If a function can have a maximum of 2 positive roots, then it could also have 0 positive roots. Also, remember that imaginary roots come from using the quadratic formula when taking the square root of a negative number. In front of that square root is a + so imaginary roots must always come in even amounts. Now, you have gathered all the information needed in order to determine the possible combinations of different types of roots. Draw a table below that summarizes the possibilities of the types of roots for h(x). Then, graph the function on the calculator and put a star by the combination that is correct. 35

38 36 For each of the functions below, use Descartes Rule of Signs to determine all of the possible combinations of positive, negative, zero, and imaginary roots that the function can have. Then, graph the function on a graphing calculator and put a star next to the combination of roots that is correct based on the graph ) ( x x x x x p ) ( 2 3 x x x x q 3. x x x x f 2 6 ) ( ) ( x x x x x g

39 Finding Rational Roots of Polynomial Functions The Rational Root Theorem f ( x) 6x 19x x 3 2 Consider the function 6. Graph the function using a graphing calculator and identify the roots from the graph. Draw what you see in the space below. The problem with this graph is that one of the roots,, seems clearly visible from the graph but the other two roots are visible but because they are not whole number values, it is impossible to tell what the values are. Thus, we have what is called the Rational Root Theorem to help us out. The Rational Root Theorem provides a list of all the POSSIBLE rational values that could be roots of the function using a manner similar to the investigation of the quadratic function at the beginning of the lesson. Rational Root Theorem: If is a polynomial with integral coefficients then is a rational root of if and only if and are relatively prime and Now, apply the Rational Root Theorem to list all of the possible values that could be roots of the 3 2 function f ( x) 6x 19x x 6. From the list above, eliminate those values that do not appear to be the values indicated on the graph. Which values from the list do appear to be roots? What can be done to verify that they are, in fact, roots of the function? Show that work below. 37

40 1. Let (a) How many roots does have? (b) Descartes Rule of Signs Test: (c) List all the possible rational roots. (d) Find the roots using the remainder theorem and synthetic division. (e) Express as a product of linear factors. 38

41 f ( x) 2x x 7x 4x Consider the function 4 whose graph is pictured below. (a) What do you notice about this graph that does not make sense based on the degree of the function? (b) What conclusion can you draw about the four roots of the function, f(x)? Explain your reasoning. Note: The goal of this lesson is to use the Rational Root Theorem to aid us in finding all of the roots of the function whether they be real or imaginary. (c) Make a list of the possible rational roots that the rational root theorem guarantees are possible. (d) From this list, which roots appear to be roots from the graph above? Perform synthetic division to verify that they are, in fact, roots. Divide f(x) by one of the associated factors. Then, divide that result by the other associated factor. (e) You have just divided a degree 4 function twice so your resulting polynomial is a. This function s roots will be the remaining two roots, which will be imaginary, of f(x). Find these imaginary roots. 39

42 Based on the results of the previous page, provide a plan on how to find all of the roots of a polynomial function. 3. Let (a) How many positive and negative roots does have? (b) Find the roots. 40

43 4. Given the polynomial, (a) find the number of possible positive roots. (b) find the number of possible negative roots. (c) find the possible rational roots. (d) find all zeros of the polynomial. (e) analyze and graph the polynomial. 41

44 5. Given the polynomial, (a) find the number of possible positive roots. (b) find the number of possible negative roots. (c) find the possible rational roots. (d) find all zeros of the polynomial. (e) analyze and graph the polynomial. 42

45 Polynomial Functions Review Problems 1. Find the zeros of the polynomial:. 2. Factor the polynomial: 3. Use synthetic division to find the quotient and remainder if is divided by. 4. List all the zeros and their multiplicities of the polynomial:. 5. Given and the fact that one of its zeros is, find all other zeros. 43

46 6. Write a polynomial with integral coefficients and of lowest possible degree that has 1 and as given zeros. 7. The table of values to the right includes points that lie on the graph of, a polynomial function. Based on the values in the table, between what two integers is there guaranteed to be a zero of the function? (A) 2 and 4 (B) 0 and 2 (C) -4 and -2 (D) -2 and 0 (E) Both B and D 8. Sketch a possible graph of a polynomial function with the given roots and multiplicity below. Then, identify the type of function. Roots: with a multiplicity of 2 with a multiplicity of 1 with a multiplicity of 1 Type of function: Equation: 44

47 9. Domain: Range: Intervals of values where is increasing: Intervals of values where is decreasing: Left End Behavior: Right End Behavior: Zeros and their multiplicity: Coordinates of point(s) of Inflection: Type of Function: Is the leading coefficient positive or negative? Possible Equation: 45

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