# Section 3-3 Approximating Real Zeros of Polynomials

Size: px
Start display at page:

Download "Section 3-3 Approximating Real Zeros of Polynomials"

Transcription

1 - 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 discussed in the preceding section are designed to find as many eact real and imaginary zeros as possible. But there are zeros that cannot be found by using these methods. For eample, the polynomial P() must have at least one real zero (Theorem in Section -). Since the only possible rational zeros are, and neither of these turns out to be a zero, P() must have at least one irrational zero. We cannot find the eact value of this zero, but it can be approimated using various well-known methods. In this section we develop two important tools for locating real zeros, the location theorem and the upper and lower bound theorem. We will see that the upper and lower bound theorem is a useful tool for approimating real zeros on a graphing utility. Net we discuss how the location theorem leads to the bisection method, a simple approimation technique that forms the basis for the zero approimation routines on most graphing utilities. Finally, we investigate some of the problems that can be encountered when a polynomial has multiple zeros. FIGURE P(). P() In this section, we restrict our attention to the real zeros of polynomials with real coefficients. Locating Real Zeros Let us return to the polynomial function P() As we have found, P() has no rational zeros and at least one irrational zero. The graph of P() is shown in Figure. Note that P() and P(). Since the graph of a polynomial function is continuous, the graph of P() must cross the ais at least once between and. This observation is the basis for Theorem and leads to a simple method for approimating zeros. THEOREM LOCATION THEOREM If f is continuous on an interval I, a and b are two numbers in I, and f(a) and f(b) are of opposite sign, then there is at least one intercept between a and b.

2 POLYNOMIAL AND RATIONAL FUNCTIONS We will find Theorem very useful when we are searching for real zeros, hence the name location theorem. It is important to remember that at least in Theorem means one or more. Notice in Figure (a) that f(), f(), and f has one zero between and. In Figure (b), f() and f(), but this time there are three zeros between and. FIGURE The location theorem. f() f() f() (a) f() (b) f() (c) f() The converse to the location theorem (Theorem ) is false; that is, if c is a zero of f, then f may or may not change sign at c. Compare Figures (a) and (c). Both functions have a zero at, but the first changes sign at and the second does not. EXAMPLE Solution FIGURE y P() 6 9. Locating Real Zeros Let P() 6 9. Use a table to locate the zeros of P() between successive integers and check graphically. We construct a table and look for sign changes [Fig. (a)]. The table in Figure (a) shows three sign changes. According to Theorem, P() must have a real zero in each of the intervals (, ), (, ), and (, ). The graph in Figure (b) confirms this. Since P() is a cubic polynomial, we have located all of its zeros. 6 (a) (b) MATCHED PROBLEM Let P() 8. Use a table to locate the zeros of P() between successive integers and check graphically. In the solution to Eample, we located three zeros in a relatively few number of steps and could stop searching because we knew that a cubic polynomial could not have more than three zeros. But what if we had not found three zeros?

3 - Approimating Real Zeros of Polynomials Some cubic polynomials have only one real zero. How can we tell when we have searched far enough? The net theorem tells us how to find upper and lower bounds for the real zeros of a polynomial. Any number that is greater than or equal to the largest zero of a polynomial is called an upper bound of the zeros of the polynomial. Similarly, any number that is less than or equal to the smallest zero of the polynomial is called a lower bound of the zeros of the polynomial. Theorem, based on the synthetic division process, enables us to determine upper and lower bounds of the real zeros of a polynomial with real coefficients. THEOREM UPPER AND LOWER BOUNDS OF REAL ZEROS Given an nth-degree polynomial P() with real coefficients, n, a n, and P() divided by r using synthetic division:. Upper Bound. If r and all numbers in the quotient row of the synthetic division, including the remainder, are nonnegative, then r is an upper bound of the real zeros of P().. Lower Bound. If r and all numbers in the quotient row of the synthetic division, including the remainder, alternate in sign, then r is a lower bound of the real zeros of P(). [Note: In the lower-bound test, if appears in one or more places in the quotient row, including the remainder, the sign in front of it can be considered either positive or negative, but not both. For eample, the numbers,, can be considered to alternate in sign, while,, cannot.] We sketch a proof of part of Theorem. The proof of part is similar, only a little more difficult. Proof If all the numbers in the quotient row of the synthetic division are nonnegative after dividing P() by r, then P() ( r)q() R where the coefficients of Q() are nonnegative and R is nonnegative. If r, then r and Q() ; hence, P() ( r)q() R Thus, P() cannot be for any greater than r, and r is an upper bound for the real zeros of P(). Theorem requires performing synthetic division repeatedly until the desired patterns occur in the quotient row. This is a simple, but tedious, operation to carry out by hand. Table shows a simple program for a graphing calculator that will perform synthetic division.

4 POLYNOMIAL AND RATIONAL FUNCTIONS T A B L E Synthetic Division on a Graphing Utility Program SYNDIV TI-8/TI-8 TI-8/TI-86 Output Eplore/Discuss (A) If you have a TI-8, TI-8, TI-8, or TI-86 graphing calculator, enter the appropriate version of SYNDIV in your calculator eactly as shown in Table. If you have some other graphing utility that can store and eecute programs, consult your manual and modify the statements in SYNDIV so that the program works on your graphing utility. (B) Store the coefficients of P() 6 9 in L (see the first line of output in Table ) and eecute the program. Type at the R? prompt and press ENTER to display the results of synthetic division with r. To continue press ENTER again. Repeat this process for r,,, and. Press QUIT at the R? prompt to terminate the program. Compare the last number in each list with the values of P() in Figure (a). EXAMPLE Solution Bounding Real Zeros Let P() 9. Find the smallest positive integer and the largest negative integer that, by Theorem, are upper and lower bounds, respectively, for the real zeros of P(). Also note the location of any zeros discovered in the process of searching for upper and lower bounds. We can use SYNDIV or hand calculations to perform synthetic division for r,,,... until the quotient row turns nonnegative; then repeat this process

5 afdbfc afdbfc FIGURE P() 9. - Approimating Real Zeros of Polynomials for r,,,... until the quotient row alternates in sign. We organize these results in the synthetic division table shown below. In a synthetic division table we dispense with writing the product of r with each coefficient in the quotient and simply list the results in the table. It is also useful to include r in the table to detect any sign changes between r and r. UB LB This quotient row is nonnegative; hence, is an upper bound (UB). This quotient row alternates in sign; hence, is a lower bound (LB). The graph of P() 9 for is shown in Figure. Theorem implies that all the real zeros of P() are between and. We can be certain that the graph does not change direction and cross the ais somewhere outside the viewing window in Figure. We also note that there is at least one zero in (, ) and at least one in (, ), as indicated by the sign changes in the values of P() shown in the last column of the synthetic division table. MATCHED PROBLEM Let P() 7. Find the smallest positive integer and the largest negative integer that, by Theorem, are upper and lower bounds, respectively, for the real zeros of P(). Also note the location of any zeros discovered in the process of searching for upper and lower bounds. EXAMPLE Approimating Real Zeros with a Graphing Utility Let P() 7 7. (A) Find the smallest positive integer multiple of and the largest negative integer multiple of that, by Theorem, are upper and lower bounds, respectively, for the real zeros of P(). (B) Approimate the real zeros of P() to two decimal places.

6 POLYNOMIAL AND RATIONAL FUNCTIONS Solutions (A) We construct a synthetic division table to search for bounds for the zeros of P(). The size of the coefficients in P() indicates that we can speed up this search by choosing larger increments between test values FIGURE P() 7 7. UB LB , 7,7 Thus, all real zeros of P() 7 7 must lie between and. (B) Graphing P() for (Fig. ) shows that P() has three zeros. The approimate values of these zeros (details omitted) are.8,.8, and.. MATCHED PROBLEM Let P() 7 7. (A) Find the smallest positive integer multiple of and the largest negative integer multiple of that, by Theorem, are upper and lower bounds, respectively, for the real zeros of P(). (B) Approimate the real zeros of P() to two decimal places. Remark One of the most frequently asked questions concerning graphing utilities is how to determine the correct viewing window. The upper and lower bound theorem provides an answer to this question for polynomial functions. As Eample illustrates, the upper and lower bound theorem and the zero approimation routine on a graphing utility are two important mathematical tools that work very well together. The Bisection Method Now that we know more about locating real zeros of a polynomial, we want to discuss a technique that can be used to approimate real zeros. Eplore/Discuss provides an introduction to the repeated systematic application of the location theorem (Theorem ) called the bisection method. This method forms the basis for the zero approimation routines in many graphing utilities. Eplore/Discuss Let P(). Since P() and P(), the location theorem implies that P() must have at least one zero in (, ). (A) Is P(.) positive or negative? Is there a zero in (,.) or in (., )?

7 - Approimating Real Zeros of Polynomials Eplore/Discuss continued (B) Let m be the midpoint of the interval from part A that contains a zero. Is P(m) positive or negative? What does this tell you about the location of the zero? (C) Eplain how this process could be used repeatedly to approimate a zero to any desired accuracy. The bisection method used to approimate real zeros is straightforward: Let P() be a polynomial with real coefficients. If P() has opposite signs at the endpoints of the interval (a, b), then a real zero r lies in this interval. We bisect this interval [find the midpoint m (a b)/], check the sign of P(m), and choose the interval (a, m) or (m, b) on which P() has opposite signs at the endpoints. We repeat this bisecting process (producing a set of nested intervals, each half the size of the preceding one and each containing the real zero r) until we get the desired decimal accuracy for the zero approimation. At any point in the process if P(m), we stop, since m is a real zero. An eample will help clarify the process. EXAMPLE Solution FIGURE 6 Nested intervals produced by the bisection method in Table. Approimating Real Zeros by Bisection For the polynomial P() 9 in Eample, we found that all the real zeros lie between and and that each of the intervals (, ) and (, ) contained at least one zero. Use bisection to approimate a real zero on the interval (, ) to one-decimal-place accuracy. We organize the results of our calculations in a table. Since the sign of P() changes at the endpoints of the interval (.6,.6), we conclude that a real zero lies in this interval and is given by r.6 to one-decimal-place accuracy (each endpoint rounds to.6)..6.6 ( (( ) ) )..7 Figure 6 illustrates the nested intervals produced by the bisection method in Table. Match each step in Table with an interval in Figure 6. Note how each interval that contains a zero gets smaller and smaller and is contained in the preceding interval that contained the zero. T A B L E Bisection Approimation Sign Change Interval (a, b) Midpoint m P(a) Sign of P P(m) P(b) (, ) (., ) (.,.7) (.,.6) (.6,.6) We stop here

8 6 POLYNOMIAL AND RATIONAL FUNCTIONS If we had wanted two-decimal-place accuracy, we would have continued the process in Table until the endpoints of a sign change interval rounded to the same two-decimal-place number. MATCHED PROBLEM Use the bisection method to approimate to one-decimal-place accuracy a zero on the interval (, ) for the polynomial in Eample. The bisection method is easy to implement on a graphing utility. Table shows a program that computes the sequence of nested intervals shown in Table. T A B L E Bisection on a Graphing Utility Program BISECT TI 8/8 TI 8/86 Output (a) (b) FIGURE 7 (c) Eplore/Discuss (A) If you have a TI-8, TI-8, TI-8, or TI-86 graphing calculator, enter the appropriate version of BISECT in your calculator eactly as shown in Table. If you have some other graphing utility that can store and eecute programs, consult your manual and modify the statements in BISECT so that the program works on your graphing utility.

9 - Approimating Real Zeros of Polynomials 7 Eplore/Discuss continued (B) To use the program, first enter P() 9 as Y in the equation editor of the graphing utility [Fig. 7(a)]. Then enter and at the prompts and press ENTER repeatedly to generate the sequence of nested intervals [Fig. 7(b)]. Press ON and select Quit to terminate eecution [Fig. 7(c)]. FIGURE 8 P() ( ) ( ).. Approimating Multiple Zeros Consider the polynomial P() ( ) ( ) which has a simple zero at, a zero of multiplicity at and a zero of multiplicity at (see Fig. 8). Notice that P() has a local maimum at and does not change sign at. Also, has a local minimum at and does not change sign there either. Both of these are zeros of even multiplicity. On the other hand, is a zero of odd multiplicity, P() does change sign at, and does not have a local etremum at. Theorem, which we state without proof, generalizes these observations. THEOREM ZEROS OF EVEN AND ODD MULTIPLICITY If P() is a polynomial with real coefficients, then. If r is a zero of odd multiplicity, then P() changes sign at r and does not have a local etremum at r.. If r is a zero of even multiplicity, then P() does not change sign at r and has a local etremum at r. Eplore/Discuss Let P() ( ) ( ). (A) Use BISECT to approimate the zeros of P(). Where does it fail to work? Why? (B) Use the zero approimation routine on your graphing utility to approimate the zeros of P(). Does this routine ever fail to work? The bisection method shown in Table requires that the function change sign at a zero to approimate that zero. Thus, this method will always fail at a zero of even multiplicity. Most graphing utilities use a more complicated routine that may or may not work at zeros of even multiplicity. For eample, the TI-8 graphing calculator was able to approimate the zero of P() ( ) ( ) at, but not the zero at. What are we to do if the zero routine fails? The ideas introduced in Theorem provide the answer:

10 8 POLYNOMIAL AND RATIONAL FUNCTIONS Zeros of even multiplicity can be approimated by using a maimum or minimum approimation routine, whichever applies. The net eample illustrates this approach. EXAMPLE Solution FIGURE 9 Zeros of P() 6 6. Approimating Multiple Zeros Let P() 6 6. Approimate the zeros of P() to two decimal places. Use maimum and minimum routines to approimate any zeros of even multiplicity. Determine the multiplicity of each zero. Eamining the graph of P(), we see that there is a zero of odd multiplicity at [Fig. 9(a)]. It appears that there may be zeros of even multiplicity near and. Using a maimum routine near [Fig. 9(b)] and a minimum routine near [Fig. 9(c)], we find that. and. are zeros of even multiplicity. Since P() is a fifth-degree polynomial, these zeros must be double zeros and must be a simple zero. (a) Simple zero (b) Double zero (c) Double zero MATCHED PROBLEM Let P() 6 7. Approimate the zeros of P() to two decimal places. Use maimum and minimum routines to approimate any zeros of even multiplicity. Determine the multiplicity of each zero. Application EXAMPLE 6 FIGURE Construction An oil tank is in the shape of a right circular cylinder with a hemisphere at each end (see Fig. ). The cylinder is inches long, and the volume of the tank is, cubic inches (approimately cubic feet). Let denote the common radius of the hemispheres and the cylinder. inches (A) Find a polynomial equation that must satisfy. (B) Approimate to one decimal place.

11 - Approimating Real Zeros of Polynomials 9 Solutions (A) If is the common radius of the hemispheres and the cylinder in inches, then ( ) ( ) ( ) Volume of tank Volume of two hemispheres Volume of cylinder,, 6 6, Multiply by /. Thus, must be a positive zero of P() 6, (B) Since the coefficients of P() are large, we use larger increments in the synthetic division table: 6,,, UB,9 6, Graphing y P() for (Fig. ), we see that. inches (to one decimal place). FIGURE P() 6,. 7, 7, MATCHED PROBLEM 6 Repeat Eample 6 if the volume of the tank is, cubic inches. Answers to Matched Problems. Intervals containing zeros: (, ), (, ), (, 6). Lower bound: ; upper bound: 6; intervals containing zeros: (, ), (, ). (A) Lower bound: ; upper bound: (B) Real zeros:.,.6, (double zero), (simple zero),.6 (double zero) 6. (A) P() 6, (B).7 inches

12 POLYNOMIAL AND RATIONAL FUNCTIONS EXERCISE - A In Problems, use the table of values for the polynomial function P to discuss the possible locations of the intercepts of the graph of y P().... P() P() respectively, for the real zeros of P(). Also note the location of any zeros between successive integers. (B) If (k, k ) is the interval determined in part A that contains the largest real zero of P(), determine the number of additional intervals required by the bisection method to obtain a one-decimal-place approimation to this zero and state the approimate value of the zero.. P() 6. P() 7. P() 8. P() 9. P() P() 9 9. P(). P(). P() P() In Problems, (A) Find the smallest positive integer and largest negative integer that, by Theorem, are upper and lower bounds, respectively, for the real zeros of P(). (B) Approimate the real zeros of each polynomial to two decimal places. In Problems 8, use a synthetic division table and Theorem to locate each real zero between successive integers.. P() 9 6. P() 9 7. P() 8. P() Find the smallest positive integer and largest negative integer that, by Theorem, are upper and lower bounds, respectively, for the real zeros of each of the polynomials given in Problems P(). P(). P() 9. P() 6 7. P(). P() B In Problems, (A) Find the smallest positive integer and largest negative integer that, by Theorem, are upper and lower bounds,. P() 8. P(). P() 7 6. P() 8 7. P() 8. P() 9. P(). P() 6 9 Problems refer to the polynomial P() ( ) ( )( ). Can the zero at be approimated by the bisection method? Eplain.. Can the zero at be approimated by the bisection method? Eplain.. Can the zero at be approimated by the bisection method? Eplain.. Which of the zeros can be approimated by a maimum approimation routine? By a minimum approimation routine? By the zero approimation routine on your graphing utility?

13 - Approimating Real Zeros of Polynomials For each polynomial P() in Problems, use a maimum or minimum routine to approimate zeros of even multiplicity and a zero approimation routine to approimate zeros of odd multiplicity, all to two decimal places. State the multiplicity of each zero.. P() P() P() P() P() P() C In Problems, (A) Find the smallest positive integer multiple of and largest negative integer multiple of that, by Theorem, are upper and lower bounds, respectively, for the real zeros of each polynomial. (B) Approimate the real zeros of each polynomial to two decimal places.. P(). P() 7 7. P() 9,. P() 7,. P(),, 6. P() 7, 7. P(),7 7,87, 8. P() 9,8,67 8, 9. P().. 9,. P() ,6, APPLICATIONS Epress the solutions to Problems 6 as the roots of a polynomial equation of the form P() and approimate these solutions to three decimal places. Use a graphing utility, if available; otherwise, use the bisection method. inches on a side, will be cut from each corner, and then the ends and sides will be folded up (see the figure). Find the value of that would result in a bo with a volume of 6 cubic inches.. Manufacturing. A bo with a hinged lid is to be made out of a piece of cardboard that measures by inches. Si squares, inches on a side, will be cut from each corner and the middle, and then the ends and sides will be folded up to form the bo and its lid (see the figure). Find the value of that would result in a bo with a volume of cubic inches. in. 8 in. in. in.. Construction. A propane gas tank is in the shape of a right circular cylinder with a hemisphere at each end (see the figure). If the overall length of the tank is feet and the volume is cubic feet, find the common radius of the hemispheres and the cylinder. feet 6. Shipping. A shipping bo is reinforced with steel bands in all three directions (see the figure). A total of. feet of steel tape is to be used, with 6 inches of waste because of a -inch overlap in each direction. If the bo has a square base and a volume of cubic feet, find its dimensions.. Geometry. Find all points on the graph of y that are unit away from the point (, ). [Hint: Use the distancebetween-two-points formula from Section -.]. Geometry. Find all points on the graph of y that are unit away from the point (, ).. Manufacturing. A bo is to be made out of a piece of cardboard that measures 8 by inches. Squares, y

### 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

### Section 1-4 Functions: Graphs and Properties

44 1 FUNCTIONS AND GRAPHS I(r). 2.7r where r represents R & D ependitures. (A) Complete the following table. Round values of I(r) to one decimal place. r (R & D) Net income I(r).66 1.2.7 1..8 1.8.99 2.1

### 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

### 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

### 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:

### Section 3-7. Marginal Analysis in Business and Economics. Marginal Cost, Revenue, and Profit. 202 Chapter 3 The Derivative

202 Chapter 3 The Derivative Section 3-7 Marginal Analysis in Business and Economics Marginal Cost, Revenue, and Profit Application Marginal Average Cost, Revenue, and Profit Marginal Cost, Revenue, and

### 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

### 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

### 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

### 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

### 3.3 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

### 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

### Section 6-3 Double-Angle and Half-Angle Identities

6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities

### Roots of equation fx are the values of x which satisfy the above expression. Also referred to as the zeros of an equation

LECTURE 20 SOLVING FOR ROOTS OF NONLINEAR EQUATIONS Consider the equation f = 0 Roots of equation f are the values of which satisfy the above epression. Also referred to as the zeros of an equation f()

### 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

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### 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

### 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

### Section 2-3 Quadratic Functions

118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the

### Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

### Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

### 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

### Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2

4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year

### SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

### 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

### FACTORING ax 2 bx c WITH a 1

296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In

### 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

### LIMITS AND CONTINUITY

LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

### Roots of Equations (Chapters 5 and 6)

Roots of Equations (Chapters 5 and 6) Problem: given f() = 0, find. In general, f() can be any function. For some forms of f(), analytical solutions are available. However, for other functions, we have

### Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

### 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

### Mathematical Modeling and Optimization Problems Answers

MATH& 141 Mathematical Modeling and Optimization Problems Answers 1. You are designing a rectangular poster which is to have 150 square inches of tet with -inch margins at the top and bottom of the poster

### When I was 3.1 POLYNOMIAL FUNCTIONS

146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

### A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

### Polynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)

Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland) POLYNOMIALS Aim To demonstrate how the TI-8 can be used

### Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

### 2-5 Rational Functions

-5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

### 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,

### D.3. Angles and Degree Measure. Review of Trigonometric Functions

APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

### Algebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED

Algebra Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED. Graph eponential functions. (Sections 7., 7.) Worksheet 6. Solve eponential growth and eponential decay problems. (Sections 7., 7.) Worksheet 8.

### 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

### Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

### Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd

5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts

### In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)

Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

### Homework 2 Solutions

Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to

### 1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names.

Pre Calculus Worksheet. 1. Which of the 1 parent functions we know from chapter 1 are power functions? List their equations and names.. Analyze each power function using the terminology from lesson 1-.

### 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

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### 9.3 OPERATIONS WITH RADICALS

9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### I think that starting

. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

### 7.2 Quadratic Equations

476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

### 7.7 Solving Rational Equations

Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

### 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

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### Section 5-9 Inverse Trigonometric Functions

46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

### A positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated

Eponents Dealing with positive and negative eponents and simplifying epressions dealing with them is simply a matter of remembering what the definition of an eponent is. division. A positive eponent means

### Lyman Memorial High School. Pre-Calculus Prerequisite Packet. Name:

Lyman Memorial High School Pre-Calculus Prerequisite Packet Name: Dear Pre-Calculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These

### Work as the Area Under a Graph of Force vs. Displacement

Work as the Area Under a Graph of vs. Displacement Situation A. Consider a situation where an object of mass, m, is lifted at constant velocity in a uniform gravitational field, g. The applied force is

### The graphs of linear functions, quadratic functions,

1949_07_ch07_p561-599.qd 7/5/06 1:39 PM Page 561 7 Polynomial and Rational Functions 7.1 Polynomial Functions 7. Graphing Polynomial Functions 7.3 Comple Numbers 7.4 Graphing Rational Functions 7.5 Equations

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### MATHCOUNTS TOOLBOX Facts, Formulas and Tricks

MATHCOUNTS TOOLBOX Facts, Formulas and Tricks MATHCOUNTS Coaching Kit 40 I. PRIME NUMBERS from 1 through 100 (1 is not prime!) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 II.

### 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

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### Sect. 1.3: Factoring

Sect. 1.3: Factoring MAT 109, Fall 2015 Tuesday, 1 September 2015 Algebraic epression review Epanding algebraic epressions Distributive property a(b + c) = a b + a c (b + c) a = b a + c a Special epansion

### Logo Symmetry Learning Task. Unit 5

Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to

### 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

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### Implicit Differentiation

Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some

### M 1310 4.1 Polynomial Functions 1

M 1310 4.1 Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a, a,..., a, a, a n n1 2 1 0, be real numbers, with a

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Business and Economic Applications

Appendi F Business and Economic Applications F1 F Business and Economic Applications Understand basic business terms and formulas, determine marginal revenues, costs and profits, find demand functions,

### Calculus 1st Semester Final Review

Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### 135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.

13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the

### 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

### 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

### 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

### 15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

### 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

### A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

### 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

### 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

### 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

### Estimating the Average Value of a Function

Estimating the Average Value of a Function Problem: Determine the average value of the function f(x) over the interval [a, b]. Strategy: Choose sample points a = x 0 < x 1 < x 2 < < x n 1 < x n = b and

### with functions, expressions and equations which follow in units 3 and 4.

Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model

### Geometry Notes VOLUME AND SURFACE AREA

Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate

### SAMPLE. Polynomial functions

Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

### C3: Functions. Learning objectives

CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

### I remember that when I

8. Airthmetic and Geometric Sequences 45 8. ARITHMETIC AND GEOMETRIC SEQUENCES Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I

### Polynomial Functions

Polynomial Functions Lessons 7-1 and 7-3 Evaluate polynomial functions and solve polynomial equations. Lessons 7- and 7-9 Graph polynomial and square root functions. Lessons 7-4, 7-5, and 7-6 Find factors

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots