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• Section 2.5 Zeros of Polynomial Functions Name______________________________________________ Section 2.5 Zeros of Polynomial Functions Objective: In this lesson you learned how to determine the number of rational and real zeros of polynomial functions, and find the zeros. Important Vocabulary Define each term or concept. Conjugates A pair of complex numbers of the form a + bi and a – bi are complex conjugates of each other. Irreducible over the reals A quadratic factor with no real zeros; also known as prime. Variation in sign Two consecutive coefficients have opposite signs. Upper bound A real number b is an upper bound for the real zeros of f if no real zeros of f are greater than b. Lower bound A real number b is a lower bound for the real zeros of f if no real zeros of f are less than b. I. The Fundamental Theorem of Algebra (Page 169) The Fundamental Theorem of Algebra guarantees that, in the complex number system, every nth-degree polynomial function has precisely n zeros. What you should learn How to use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions Example 1: How many zeros does the polynomial function f ( x) = 5 − 2 x 2 + x 3 − 12 x 5 have? 5 The Linear Factorization Theorem states that . . . if f(x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors f(x) = an(x – c1)( x – c2) . . . (x – cn) where c1, c2, . . . , cn are complex numbers. II. The Rational Zero Test (Pages 170−172) Describe the purpose of the Rational Zero Test. The Rational Zero Test relates the possible rational zeros of a polynomial (having integer coefficients) to the leading coefficient and to the constant term of the polynomial. Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to find rational zeros of polynomial functions 45 46 Chapter 2 • Polynomial and Rational Functions State the Rational Zero Test. If the polynomial f(x) = anxn + an – 1xn – 1 + . . . + a2x2 + a1x + a0 has integer coefficients, every rational zero of f has the form: rational zero = p/q, where p and q have no common factors other than 1, and p = a factor of the constant term a0, and q = a factor of the leading coefficient an. To use the Rational Zero Test, . . . first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Then use trial and error to determine which of these possible rational zeros, if any, are actual zeros of the polynomial. Example 2: List the possible rational zeros of the polynomial function f ( x) = 3 x 5 + x 4 + 4 x 3 − 2 x 2 + 8 x − 5 . ± 1, ± 5, ± 1/3, ± 5/3 Some strategies that can be used to shorten the search for actual zeros among a list of possible rational zeros include . . . (1) a programmable calculator can be used to speed up the calculations; (2) a graph, drawn either by hand or with a graphing utility, can give a good estimate of the locations of the zeros; (3) the Intermediate Value Theorem along with a table generated by a graphing utility can give approximations of zeros; and (4) synthetic division can be used to test possible rational zeros. III. Conjugate Pairs (Page 173) Let f(x) be a polynomial function that has real coefficients. If What you should learn How to find conjugate pairs of complex zeros a + bi (where b ≠ 0) is a zero of the function, then we know that a − bi is also a zero of the function. Example 3: Give the complex conjugate of 3 − 7i. 3 + 7i Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 2.5 • Zeros of Polynomial Functions Name______________________________________________ IV. Factoring a Polynomial (Pages 173−175) To write a polynomial of degree n > 0 with real coefficients as a What you should learn How to find zeros of polynomials by factoring product without complex factors, write the polynomial as . . . the product of linear and/or quadratic factors with real coefficients, where the quadratic factors have no real zeros. Example 4: Write the polynomial function f ( x) = x 4 + 5 x 2 − 36 as the product of linear factors, and list all of its zeros. f(x) = (x + 2)(x − 2)(x + 3i)(x − 3i) Zeros: − 2, 2, − 3i, 3i Explain why a graph cannot be used to locate complex zeros. Real zeros are the only zeros that appear as x-intercepts on a graph. A polynomial function’s complex zeros must be found algebraically. V. Other Tests for Zeros of Polynomials (Pages 176−178) Descartes’s Rule of Signs sheds more light on the number of real zeros a polynomial function can have. State Descartes’s Rule of Signs. Let f(x) = anxn + an – 1xn – 1 + . . . + a2x2 + a1x + a0 be a polynomial with real coefficients and a0 ≠ 0. 1. The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer. 2. The number of negative real zeros of f is either equal to the number of variations in sign of f(– x) or less than that number by an even integer. When using Descartes’s Rule of Signs, a zero of multiplicity k should be counted as k zeros. Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to use Descartes’s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials 47 48 Chapter 2 • Polynomial and Rational Functions Example 5: Find the number of variations in sign in f ( x) = 2 x 6 + 3x 5 − x 4 − 9 x 3 + x 2 + 5 x − 7 , as well as the number of variations of sign in f(− x). Then discuss the possible numbers of positive real zeros and the possible number of negative real zeros of this function. 3 changes in sign for f(x); 3 changes in sign for f(− x) f(x) has either 3 or 1 positive real zeros, and either 3 or 1 negative real zeros. State the Upper and Lower Bound Rules. Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x − c, using synthetic division. 1. If c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f. 2. If c < 0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f. Explain how the Upper and Lower Bound Rules can be useful in the search for the real zeros of a polynomial function. Explanations will vary. For instance, suppose you are checking a list of possible rational zeros. When checking the possible rational zero 2 with synthetic division, each number in the last row is positive or zero. Then you need not check any of the other possible rational zeros that are greater than 2 and can concentrate on checking only values less than 2. Homework Assignment Page(s) Exercises Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved.