Download presentation

Presentation is loading. Please wait.

1
**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

2
**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

3
**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

4
**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

5
**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

6
**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)

7
**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

8
**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)

9
**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)

10
**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)

11
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)

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google