# Digital Lesson Polynomial Functions.

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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 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. x 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 – 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 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 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

Use your graphing calculator and division to find out the zero that is not an integer for the function f(x)=2x4 – 17x3 + 35x2 + 9x – 45. 2x4 – 17x3 + 35x2 + 9x – 45 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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

1. Divide using long division.

1. Divide using synthetic division.

8.Divide using long division

Use synthetic division to find each function value
Use synthetic division to find each function value. Use a graphing utility to verify. 31. a. f(1) b. f(-2) c. f(1/2) d. f(8) Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all the real zeros of the function. x=2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.