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**Factoring Polynomials**

Divide: (x2 – 5x – 9) ÷ (x – 3) (x ¹ 3) 3 1 – 5 – 9 3 3 – 6 1 – 2 – 15 Ans: x – 2 R – 15 Sub x = 3 into: x2 – 5x – 9 = 32 – 5(3) – 9 = – 15

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**Remainder Theorem In order to determine the remainder when**

P(x) is divided by (x – k), replace x by k. P(k) = r In order to determine the remainder when replace x by P(x) is divided by (jx – k), P( ) = r

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**Example 1: Determine the remainder when**

2x3 – 5x2 + 2x – 4 is divided by (x + 2). P(x) = 2x3 – 5x x – 4 P(–2) = 2(–2)3 – 5(–2)2 + 2(–2) – 4 P(–2) = 2(–8) – 5(4) – – 4 P(–2) = – 16 – 20 – – 4 P(–2) = – 44 The remainder is – 44

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**Example 2: Determine the remainder when**

4x2 + 2x – 3 is divided by (2x – 1). P(x) = 4x x – 3 P( ) = 4( )2 + 2( ) – 3 P( ) = 4( ) – 3 P( ) = – 3 P( ) = – 1 \ The remainder is –1

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**Example 1: Factor x3 – 2x2 – 5x + 6**

Factor Theorem A polynomial P(x) has (x – k) as factor if and only if P(k) = 0. Example 1: Factor x3 – 2x2 – 5x + 6 P(x) = x3 – 2x2 – 5x + 6 let x = 1 P(1) = (1)3 – 2(1)2 – 5(1) + 6 P(1) = 1 – 2 – P(1) = 0 \ x – 1 is a factor

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**To find another factor, divide**

(x3 – 2x2 – 5x + 6) by (x – 1) 1 – 2 – 1 1 – 1 – 6 1 – 1 – 6 (x2 – x – 6) = (x – 3)(x + 2) x3 – 2x2 – 5x + 6 = (x – 1)(x – 3)(x + 2)

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**Example 2: Factor 2x3 – 3x2 – 3x + 2**

P(x) = 2x3 – 3x2 – 3x + 2 let x = 1 P(1) = 2(1)3 – 3(1)2 – 3(1) + 2 P(1) = 2 – 3 – P(1) = – 2 let x = –1 P(–1) = 2(–1)3 – 3(–1)2 – 3(–1) + 2 P(–1) = –2 – P(–1) = 0 \ (x + 1) is a factor

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**Divide (2x3 – 3x2 – 3x + 2) by (x + 1)**

2 – 3 – –1 – 2 5 – 2 now factor this 2 – 5 2 2x2 – 5x + 2 = (2x – 1)(x – 2) \ 2x3 – 3x2 – 3x + 2 = (x + 1)(2x – 1)(x – 2)

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**Example: Factor the following:**

Factoring by Grouping Example: Factor the following: 4x3 – 8x2 – 9x + 18 group them in pairs = 4x2(x – 2) – 9(x – 2) common factor = (x – 2)(4x2 – 9) difference of squares = (x – 2)(2x – 3)(2x + 3)

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**The sum and difference of cubes:**

In general: (x3 + y3) = (x + y) (x2 – xy + y2) 1. cube root of each term. 2. square the first root. 3. product of the roots (opposite sign) 4. square of the second root. (x3 – y3) = (x – y)(x2 + xy + y2)

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Factor the following: (x3 + 8) = (x + 2) (x2 – 2x + 4) (x3 – 64) = (x – 4) (x2 + 4x + 16) (x3 + 27) = (x + 3) (x2 – 3x + 9) (x3 – 125) = (x – 5) (x2 + 5x + 25)

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