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Chapter 2 Polynomial and Rational Functions

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Warm Up 2.3 An object is launched at 19.6 meters per second from a 58.8 -meter tall platform. The equation for the object’s height s at time t seconds after launch is where s is in meters. When does the object strike the ground? 2

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2.3 Real Zeros of Polynomial Functions Objectives: Use long division to divide polynomials by other polynomials. Use synthetic division to divide polynomials by binomials of the form ( x – k ). Use the Remainder and Factor Theorems. Use the Rational Zero Test to determine possible rational zeros of polynomial functions. Use Descartes’s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. 3

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Consider This …. Consider the graph of the function One zero is x = ___________. The linear factor is ______________. Use the linear factor to produce a complete factorization of the function. That is, factor the function completely. Then find the zeros. 4

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Polynomial Long Division Divide x 2 +3x +5 by x + 1. Identify the dividend, divisor, quotient, and remainder. 5

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Notes on Division Before performing division, be sure to: 1. Write the dividend and divisor in descending powers of the variable, that is, in standard polynomial form. 2. Insert zeros for any missing powers of the variable. 6

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Examples Use polynomial long division to solve. 7

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Synthetic Division A shortcut for polynomial long division. Divisor must linear, that is, x – k. Example: Divide x 4 – 10x 2 – 2x +4 by x + 3 8

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Examples Use synthetic division to solve. 9

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The Remainder Theorem If a polynomial f (x) is divided by x – k, the remainder is r = f (k). To evaluate a polynomial function at a value x = k, divide the polynomial by x – k. The remainder will equal f (k). 10

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Examples Evaluate the function at x = 2. Evaluate the function at x = –4. 11

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The Factor Theorem A polynomial f (x) has a factor (x – k) if and only if f (k) = 0. We can demonstrate that some (x – k) is a factor of f (x) by dividing and showing that the remainder is equal to zero. 12

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Example Show that (x – 2) and (x + 3) are factors of Then find the remaining factors of the function. 13

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Real Zeros The real zeros of a polynomial can be rational or irrational. If a polynomial has integer coefficients, we can find the potential rational zeros by using the factors of the leading coefficient a n and the constant term a 0. 14

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The Rational Zero Test 15

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Using the Rational Zero Test Find a 0 and a n. List all factors (positive and negative) of a 0 and a n. Write the possible rational zeros as a quotient: Check to see which of these numbers are zeros by using synthetic division. Continue factoring. 16

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Helpful Hint Graph the function on your calculator FIRST to eliminate unlikely possible zeros. 17

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Example Find the rational zeros of What is the constant term? What is the leading coefficient? What are the factors of each? Write the possible rational zeros. Use synthetic division to find the first zero. Factor completely. 18

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Example Find the rational zeros of What is the constant term? What is the leading coefficient? What are the factors of each? Write the possible rational zeros. Since the list is so long, use calculator to narrow down the search. Use synthetic division to find the first zero. Factor completely. 19

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Homework 2.2 Worksheet 2.2 20

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