1. Warm-Up #22
(3m + 4)(-m + 2)
Factor out ( 96𝑥 3 − 48𝑥 2 +60𝑥)

2. Factoring Binomials

3. Definitions
To factor an expression means to write an equivalent expression that is a product
To factor a polynomial means to write the polynomial as a product of other polynomials
A factor that cannot be factored further is said to be a prime factor (prime polynomial)
A polynomial is factored completely if it is written as a product of prime polynomials

4. Distributive Property using FOIL
A(B + C) = AB + AC
A(B – C) = AB – AC
Example: 5x(6x + 3)

5. Distributive Property
Example: (x+2)(6x + 3)

6. Distributive Property
Example: 5x(6x + 3)(-2x – 1)

7. Factor the expression
30𝑥 2 +15𝑥
5x(6x + 3)

8. Ex: Factor 6a5 – 3a3 – 2a2 Recall: GCF(6a5,3a3,2a2)= a21
6a5 – 3a3 – 2a2 = a2( )
6a3 3a 2 a2 a2 a2
 6a5 – 3a3 – 2a2 = a2(6a3 – 3a – 2)

9. Ex: Factor x(a + b) – 2(a + b)
Always ask first if there is common factor the terms share . . .
x(a + b) – 2(a + b)
Each term has factor (a + b)
 x(a + b) – 2(a + b)
= (a + b)( – ) x 2
(a + b)
(a + b)
 x(a + b) – 2(a + b) = (a + b)(x – 2)

10. Ex: Factor a(x – 2) + 2(2 – x)
As with the previous example, is there a common factor among the terms?
Well, kind of x – 2 is close to 2 - x Hum . . .
Recall: (-1)(x – 2) =
- x + 2 = 2 – x
 a(x – 2) + 2(2 – x) =
a(x – 2) + 2((-1)(x – 2))
= a(x – 2) + (– 2)(x – 2)
= a(x – 2) – 2(x – 2)
 a(x – 2) – 2(x – 2) =
(x – 2)( – ) a 2
(x – 2)
(x – 2)

11. Ex: Factor b(a – 7) – 3(7 – a) Common factor among the terms?
Well, kind of a – 7 is close to 7 - a
Recall: (-1)(a – 7) =
- a + 7 = 7 – a
 b(a – 7) – 3(7 – a) =
b(a – 7) – 3((-1)(a – 7))
= b(a – 7) + 3(a – 7)
= b(a – 7) +3(a – 7)
 b(a – 7) + 3(a – 7) =
(a – 7)( ) b 3
(a – 7)
(a – 7)

12. Special Binomials Is the polynomial a difference of squares?
a2 – b2 = (a – b)(a + b)
Is the trinomial a perfect-square trinomial?
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2

13. Special Binomials Is the polynomial a difference of squares?
a2 – b2 = (a – b)(a + b)
LIST ALL OF THE PERFECT SQUARES:

14. Ex: Factor x2 – 4
Notice the terms are both perfect squares
and we have a difference
 difference of squares
x2 = (x)2
4 = (2)2
 x2 – 4 = (x)2 – (2)2
= (x – 2)(x + 2)
a2 – b2
= (a – b)(a + b)
factors as

15. Example 1: 𝑥 2 −36
Example 2: 𝑥 2 −125

16. Ex: Factor 9p2 – 16
Notice the terms are both perfect squares
and we have a difference
 difference of squares
9p2 = (3p)2
16 = (4)2
 9a2 – 16 = (3p)2 – (4)2
= (3p – 4)(3p + 4)
a2 – b2
= (a – b)(a + b)
factors as

17. Example 1: 16𝑥 2 −25
Example 2: 17𝑥 2 −49

18. Ex: Factor y6 – 25
Notice the terms are both perfect squares
and we have a difference
 difference of squares
y6 = (y3)2
25 = (5)2
 y6 – 25 = (y3)2 – (5)2
= (y3 – 5)(y3 + 5)
a2 – b2
= (a – b)(a + b)
factors as

19. Example 1: 𝑚 8 −36
Example 2: 𝑥 6 −64

20. Ex: Factor 81 – x2y2
Notice the terms are both perfect squares
and we have a difference
 difference of squares
81 = (9)2
x2y2 = (xy)2
 81 – x2y2 = (9)2 – (xy)2
= (9 – xy)(9 + xy)
a2 – b2
= (a – b)(a + b)
factors as

21. Example 1: −36 + 𝑟 2 𝑚 2
Example 2: −125 + 𝑛 4 𝑥 2