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What we will learn today…

How to divide polynomials and relate the result to the remainder and factor theorems

How to use polynomial division

Objective: 6.5 The Remainder and Factor Theorems

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Dividing Polynomials1. 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:

)(

)()(

)(

)(

xd

xrxq

xd

xf

Objective: 6.5 The Remainder and Factor Theorems

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How Do We Do This Division? Long Division!Divide 2x4 + 3x3 + 5x – 1 by x2 – 2x + 2

Objective: 6.5 The Remainder and Factor Theorems

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The Answer We write the answer as:

Objective: 6.5 The Remainder and Factor Theorems

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You Try Divide: y4 + 2y2 – y + 5 by y2 – y + 1

Objective: 6.5 The Remainder and Factor Theorems

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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.

Objective: 6.5 The Remainder and Factor Theorems

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Synthetic Division Divide x3 + 2x2 – 6x – 9 by x – 2

You Try! Divide the polynomial by x + 3

Objective: 6.5 The Remainder and Factor Theorems

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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”.

Objective: 6.5 The Remainder and Factor Theorems

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Using Synthetic Substitution Use synthetic substitution to find the

factors of: f(x) = 2x3 + 11x2 + 18x + 9 given that f(-3) = 0.

Objective: 6.5 The Remainder and Factor Theorems

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Finding Zeros of a Polynomial Function We can use synthetic division to find the

zeros of a function. Example: One zero of f(x) = x3 – 2x2 – 9x + 18 is

x=2. Find the other zeros of the function.

Objective: 6.5 The Remainder and Factor Theorems

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You Try One zero of f(x) = x3 + 6x2 + 3x – 10 is x=-5. Find the other zeros of the function.

Objective: 6.5 The Remainder and Factor Theorems

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Homework Page 356, 17, 23, 27, 35, 39, 41, 49, 53,

55, 59