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)