Digital Lesson Polynomial Functions

A polynomial function is a function of the form where n is a nonnegative integer and each ai (i = 0, , n) is a real number. The polynomial function has a leading coefficient an and degree n. Examples: Find the leading coefficient and degree of each polynomial function. Polynomial Function Leading Coefficient Degree – 2 5 1 3 14 0 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Polynomial Function

A polynomial function of degree n has at most n zeros. A real number a is a zero of a function y = f (x) if and only if f (a) = 0. Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. x = a is a zero of f. 2. x = a is a solution of the polynomial equation f (x) = 0. 3. (x – a) is a factor of the polynomial f (x). 4. (a, 0) is an x-intercept of the graph of y = f (x). A polynomial function of degree n has at most n zeros. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Zeros of a Function

Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2. Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2). y x –2 2 f (x) = x4 – x3 – 2x2 The real zeros are x = –1, x = 0, and x = 2. (–1, 0) (0, 0) These correspond to the x-intercepts. (2, 0) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Real Zeros

Dividing Polynomials Example: Divide x2 + 3x – 2 by x – 1 and check the answer. 1. x + 2 2. x2 + x 3. 2x – 2 2x + 2 4. – 4 5. remainder 6. Answer: x + 2 + – 4 Check: (x + 2) quotient (x + 1) divisor + (– 4) remainder = x2 + 3x – 2 dividend Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Dividing Polynomials

Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division. Synthetic division is a shorter method of dividing polynomials. This method can be used only when the divisor is of the form x – a. It uses the coefficients of each term in the dividend. Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division. Since the divisor is x – 2, a = 2. value of a coefficients of the dividend 1. Bring down 3 2 3 2 – 1 2. (2 • 3) = 6 3. (2 + 6) = 8 6 16 4. (2 • 8) = 16 3 8 15 5. (–1 + 16) = 15 coefficients of quotient remainder 3x + 8 Answer: 15 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Synthetic Division

The remainder is 68 at x = 3, so f (3) = 68. Remainder Theorem: The remainder of the division of a polynomial f (x) by x – k is f (k). Example: Using the remainder theorem, evaluate f(x) = x 4 – 4x – 1 when x = 3. value of x 3 1 0 0 – 4 – 1 3 9 27 69 1 3 9 23 68 The remainder is 68 at x = 3, so f (3) = 68. You can check this using substitution: f(3) = (3)4 – 4(3) – 1 = 68. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Remainder Theorem

The remainders of 0 indicate that (x + 2) and (x – 1) are factors. Factor Theorem: A polynomial f(x) has a factor (x – k) if and only if f(k) = 0. Example: Show that (x + 2) and (x – 1) are factors of f(x) = 2x 3 + x2 – 5x + 2. – 2 2 1 – 5 2 1 2 – 3 1 – 4 6 – 2 2 – 1 2 – 3 1 2 – 1 The remainders of 0 indicate that (x + 2) and (x – 1) are factors. The complete factorization of f is (x + 2)(x – 1)(2x – 1). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Factor Theorem

where p and q have no common factors other than 1. Rational Zero Test: If a polynomial f(x) has integer coefficients, every rational zero of f has the form where p and q have no common factors other than 1. p is a factor of the constant term. q is a factor of the leading coefficient. Example: Find the rational zeros of f(x) = x3 + 3x2 – x – 3. q = 1 p = – 3 The possible rational zeros are ±1 and ±3. Synthetic division shows that the factors of f are (x + 3), (x + 1), and (x – 1). The zeros of f are – 3, – 1, and 1. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Rational Zero Test

Descartes’s Rule of Signs Descartes’s Rule of Signs: If f(x) is a polynomial with real coefficients and a nonzero constant term, The number of positive real zeros of f is equal to the number of variations in sign of f(x) or less than that number by an even integer. The number of negative real zeros of f is equal to the number of variations in sign of f(–x) or less than that number by an even integer. A variation in sign means that two consecutive, nonzero coefficients have opposite signs. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Descartes’s Rule of Signs

Example: Descartes’s Rule of Signs Example: Use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros of f(x) = 2x4 – 17x3 + 35x2 + 9x – 45. The polynomial has three variations in sign. + to – + to – f(x) = 2x4 – 17x3 + 35x2 + 9x – 45 – to + f(x) has either three positive real zeros or one positive real zero. f(– x) = 2(– x)4 – 17(– x)3 + 35(– x)2 + 9(– x) – 45 =2x4 + 17x3 + 35x2 – 9x – 45 One change in sign f(x) has one negative real zero. f(x) = 2x4 – 17x3 + 35x2 + 9x – 45 = (x + 1)(2x – 3)(x – 3)(x – 5). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Descartes’s Rule of Signs

Graphing Utility: Finding Roots Graphing Utility: Find the zeros of f(x) = 2x3 + x2 – 5x + 2. – 10 10 Calc Menu: The zeros of f(x) are x = – 2, x = 0.5, and x = 1. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graphing Utility: Finding Roots