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Published byHugo Harrell Modified over 6 years ago

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1 What we will learn today… How to divide polynomials and relate the result to the remainder and factor theorems How to use polynomial division

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Objective: 6.5 The Remainder and Factor Theorems 2 Dividing Polynomials 1. When you divide a polynomial, f(x) by a divisor, d(x), you get a quotient polynomial, q(x) and a remainder polynomial, r(x). We can write:

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Objective: 6.5 The Remainder and Factor Theorems 3 How Do We Do This Division? Long Division! Divide 2x 4 + 3x 3 + 5x – 1 by x 2 – 2x + 2

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Objective: 6.5 The Remainder and Factor Theorems 4 The Answer We write the answer as:

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Objective: 6.5 The Remainder and Factor Theorems 5 You Try Divide: y 4 + 2y 2 – y + 5 by y 2 – y + 1

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Objective: 6.5 The Remainder and Factor Theorems 6 Remainder Theorem If a polynomial f(x) is divided by x – k, then the remainder is r = f(k). For instance if the remainder after dividing a polynomial by x-2 is 15, f(2) would also be 15.

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Objective: 6.5 The Remainder and Factor Theorems 7 Synthetic Division Divide x 3 + 2x 2 – 6x – 9 by x – 2 You Try! Divide the polynomial by x + 3

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Objective: 6.5 The Remainder and Factor Theorems 8 Factor Theorem A polynomial f(x) has a factor x – k if and only if f(k) = 0 (no remainder). A number is called a zero of a function when it causes the function to evaluate to (or equal) zero. These also happen to be the “solutions”.

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Objective: 6.5 The Remainder and Factor Theorems 9 Using Synthetic Substitution Use synthetic substitution to find the factors of: f(x) = 2x 3 + 11x 2 + 18x + 9 given that f(-3) = 0.

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Objective: 6.5 The Remainder and Factor Theorems 10 Finding Zeros of a Polynomial Function We can use synthetic division to find the zeros of a function. Example: One zero of f(x) = x 3 – 2x 2 – 9x + 18 is x=2. Find the other zeros of the function.

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Objective: 6.5 The Remainder and Factor Theorems 11 You Try One zero of f(x) = x 3 + 6x 2 + 3x – 10 is x=-5. Find the other zeros of the function.

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Objective: 6.5 The Remainder and Factor Theorems 12 Homework Page 356, 17, 23, 27, 35, 39, 41, 49, 53, 55, 59

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