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2.3 Real Zeros of Polynomial Functions 2015 Digital Lesson

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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.

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Objective To find real zeros of polynomial functions using various methods. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

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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).

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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

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You try: Divide by using long division. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6

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Try another one: Divideusing long division. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

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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

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1. Divideusing synthetic division. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

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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).

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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.

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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

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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

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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

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Use the graphing calculator to find the exact value of all real zeros. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

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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

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Homework: Pg. 123 1-33 EOO, 35-43 odd,49-55 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17

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HWQ practice Solve by dividing out a zero and factoring the resulting quadratic ( or use quadratic formula): Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18

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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

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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

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Homework: 2.1/2.3 Review Worksheet Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21

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2.3 Real Zeros of Polynomial Functions Day 2 – More problems Digital Lesson

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