# Polynomials Classwork

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Polynomials Classwork

2

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

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

### In this lesson you will learn to find zeros of polynomial functions that are not factorable.

2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### 5.1 The Remainder and Factor Theorems; Synthetic Division

5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials

### SOLVING POLYNOMIAL EQUATIONS

C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

### Polynomial and Rational Functions

Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

### College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran

College Algebra - MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or x-intercept) of a polynomial is identical to the process of factoring a polynomial.

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### 3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions P(x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

### Polynomial Expressions and Equations

Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

### Solutions to Self-Test for Chapter 4 c4sts - p1

Solutions to Self-Test for Chapter 4 c4sts - p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a

### 3.6 The Real Zeros of a Polynomial Function

SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

2.5 ZEROS OF POLYNOMIAL FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions.

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### 2.4 Real Zeros of Polynomial Functions

SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower

### MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

### Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

### Algebra Tiles Activity 1: Adding Integers

Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Calculus Card Matching

Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

### Polynomials and Factoring

Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### 3.2 The Factor Theorem and The Remainder Theorem

3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

### Roots of Polynomials

Roots of Polynomials (Com S 477/577 Notes) Yan-Bin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x

### 8 Polynomials Worksheet

8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### Mth 95 Module 2 Spring 2014

Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

### Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial

### 55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### Factoring Trinomials: The ac Method

6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

### Factoring Polynomials

Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations

### i is a root of the quadratic equation.

13 14 SEMESTER EXAMS 1. This question assesses the student s understanding of a quadratic function written in vertex form. y a x h k where the vertex has the coordinates V h, k a) The leading coefficient

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

### UNIT TWO POLYNOMIALS MATH 421A 22 HOURS. Revised May 2, 00

UNIT TWO POLYNOMIALS MATH 421A 22 HOURS Revised May 2, 00 38 UNIT 2: POLYNOMIALS Previous Knowledge: With the implementation of APEF Mathematics at the intermediate level, students should be able to: -

### 0.4 FACTORING POLYNOMIALS

36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### 3-17 15-25 5 15-10 25 3-2 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true

Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

### Situation: Dividing Linear Expressions

Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product

### A. Factoring out the Greatest Common Factor.

DETAILED SOLUTIONS AND CONCEPTS - FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Polynomials and Polynomial Functions

Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions Overview Number of instruction days: 13-15 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Prove

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Unit 3 Polynomials Study Guide

Unit Polynomials Study Guide 7-5 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

### 2.5 Zeros of a Polynomial Functions

.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### UNIT 3: POLYNOMIALS AND ALGEBRAIC FRACTIONS. A polynomial is an algebraic expression that consists of a sum of several monomials. x n 1...

UNIT 3: POLYNOMIALS AND ALGEBRAIC FRACTIONS. Polynomials: A polynomial is an algebraic expression that consists of a sum of several monomials. Remember that a monomial is an algebraic expression as ax

### Derivatives and Graphs. Review of basic rules: We have already discussed the Power Rule.

Derivatives and Graphs Review of basic rules: We have already discussed the Power Rule. Product Rule: If y = f (x)g(x) dy dx = Proof by first principles: Quotient Rule: If y = f (x) g(x) dy dx = Proof,

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

### Mathematics. GSE Algebra II/Advanced Algebra Unit 3: Polynomial Functions

Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Algebra II/Advanced Algebra Unit 3: Polynomial Functions These materials are for nonprofit educational purposes only. Any other use

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### Name: where Nx ( ) and Dx ( ) are the numerator and

Oblique and Non-linear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m

### Developmental Math Course Outcomes and Objectives

Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring

Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### The Mean Value Theorem

The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers

### Section 3-3 Approximating Real Zeros of Polynomials

- Approimating Real Zeros of Polynomials 9 Section - Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### Solving Quadratic Equations by Completing the Square

9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative