1. Polynomial Functions Digital Lesson

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

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Zeros of a Function A real number a is a zero of a function y = f (x) if and only if f (a) = 0. A polynomial function of degree n has at most n zeros. 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).

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 y x –2 2 Example: Real Zeros Example: Find all the real zeros of f (x) = x 4 – x 3 – 2x 2. Factor completely: f (x) = x 4 – x 3 – 2x 2 = x 2 (x + 1)(x – 2). The real zeros are x = –1, x = 0, and x = 2. These correspond to the x-intercepts. f (x) = x 4 – x 3 – 2x 2 (–1, 0) (0, 0) (2, 0)

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 + 2 Dividing Polynomials Example: Divide x 2 + 3x – 2 by x – 1 and check the answer. x x 2 + x 2x2x– 2 2x + 2 – 4– 4 remainder Check: 1. 2. 3. 4. 5. 6. (x + 2) quotient (x + 1) divisor + (– 4) remainder = x 2 + 3x – 2 dividend Answer: x + 2 + – 4– 4 Dividing Polynomials

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 16 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 3x 2 + 2x – 1 by x – 2 using synthetic division. 3 2 – 1 2 Since the divisor is x – 2, a = 2. 3 1. Bring down 3 2. (2 3) = 6 6 815 3. (2 + 6) = 8 4. (2 8) = 16 5. (–1 + 16) = 15 coefficients of quotient remainder value of a coefficients of the dividend 3x + 8Answer: 15

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Remainder Theorem 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. 9 1 0 0 – 4 – 1 3 1 3 39 6927 2368 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. value of x

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Factor Theorem 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 + x 2 – 5x + 2. 6 2 1 – 5 2 – 2 2 – 4 – 31 – 2 0 The remainders of 0 indicate that (x + 2) and (x – 1) are factors. – 1 2 – 3 1 1 2 2 – 10 The complete factorization of f is (x + 2)(x – 1)(2x – 1).

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Rational Zero Test 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. Example: Find the rational zeros of f(x) = x 3 + 3x 2 – x – 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). p is a factor of the constant term. q is a factor of the leading coefficient. q = 1 p = – 3 The zeros of f are – 3, – 1, and 1.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Descartes’s Rule of Signs A variation in sign means that two consecutive, nonzero coefficients have opposite signs. Descartes’s Rule of Signs: If f(x) is a polynomial with real coefficients and a nonzero constant term, 1.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. 2.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.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Descartes’s Rule of Signs The polynomial has three variations in sign. Example: Use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros of f(x) = 2x 4 – 17x 3 + 35x 2 + 9x – 45. f(x) = 2x 4 – 17x 3 + 35x 2 + 9x – 45 + to – – to + + to – f(– x) = 2(– x) 4 – 17(– x) 3 + 35(– x) 2 + 9(– x) – 45 =2x 4 + 17x 3 + 35x 2 – 9x – 45 f(x) has either three positive real zeros or one positive real zero. f(x) = 2x 4 – 17x 3 + 35x 2 + 9x – 45 = (x + 1)(2x – 3)(x – 3)(x – 5). f(x) has one negative real zero. One change in sign

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Graphing Utility: Finding Roots Graphing Utility: Find the zeros of f(x) = 2x 3 + x 2 – 5x + 2. Calc Menu: The zeros of f(x) are x = – 2, x = 0.5, and x = 1. – 10 10 – 10