 # 2.3 Real Zeros of Polynomial Functions 2015 Digital Lesson.

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2.3 Real Zeros of Polynomial Functions 2015 Digital Lesson

Warm-up Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Find the zeros of the function and Sketch a graph with those zeros and correct end behavior.

4 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).

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: (x + 2) quotient (x + 1) divisor + (– 4) remainder = x 2 + 3x – 2 dividend Answer: x + 2 + – 4– 4 Dividing Polynomials

8 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 6 815 coefficients of quotient remainder value of a coefficients of the dividend 3x + 8Answer: 15

10 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).

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 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.

Find all zeros of the function given x+2 is a factor. f(x) = 2x 3 + x 2 – 5x + 2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 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

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. 14

Verify that (x+2) and (x-4) are factors of the function and find any remaining factors to write the complete factorization of f. List all real zeros. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

Use the graphing calculator to find the any exact zeros. Then use synthetic division to break down the function to find the other zeros. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19

Find all real solutions of the polynomial. Use the rational root test, or graphing calculator. Extra credit to anyone who can factor the polynomial by hand. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20