1. 8-7 Factoring Special Cases
Hubarth
Algebra

2. Perfect Squares Trinomials
For every real number a and b:
𝑎 2 +2𝑎𝑏+ 𝑏 2 = 𝑎+𝑏 𝑎+𝑏 =(𝑎+ 𝑏) 2
𝑎 2 −2𝑎𝑏+ 𝑏 2 = 𝑎−𝑏 𝑎−𝑏 =(𝑎− 𝑏) 2
EXAMPLES 𝑥 2 +10𝑥+25= 𝑥+5 𝑥+5 =(𝑥+5 ) 2
𝑥 2 −10𝑥+25= 𝑥−5 𝑥−5 =(𝑥−5 ) 2
To recognize perfect square the first and last terms must be perfect squares. The middle term
will be two times the product of the first and last terms square root.

3. Ex 1 Factoring Perfect Squares
𝑥 2 is a perfect square
16 is a perfect square
the square root of 𝑥 2 is x
the square root of 16 is 4
2(4)(x) = 8x
our middle term
Our factors are
(x + 4)(x + 4) = (𝑥+4) 2

4. Ex 2 Factoring Perfect Squares
9 𝑔 2 is perfect square
4 is a perfect square
9 𝑔 2 =3𝑔
4 = 2
2 3𝑔 2 =12𝑔
Because the middle term is negative the two factors will be negative.
3𝑔−2 3𝑔−2 = (3𝑔−2) 2

5. Difference of Squares
For every real number a and b:
𝑎 2 − 𝑏 2 =(𝑎+𝑏)(𝑎−𝑏)
EXAMPLES 𝑥 2 −81=(𝑥−9)(𝑥+9)
16𝑥 2 −49=(4𝑥−7)(4𝑥+7)
Both terms need to be perfect squares. The factors are the square roots of each term

6. Ex 3 Factor the Difference of Squares
𝑥−4 (𝑥+4)
Ex 4 Factor the Difference of Squares
Factor 4𝑥 2 −121
(2𝑥+11)(2𝑥−11)

7. Ex 5 Factoring Out a Common Factor
Find the common factor of each term. 10 goes into both terms.
10( 𝑥 2 −4)
Now, factor the difference of squares.
10(𝑥+2)(𝑥−2)

8. Practice
Factor each expression
1. 𝑛 2 −16𝑛 𝑥 2 +4𝑥 𝑡 2 −36𝑡+81
4. 4𝑥 2 − 𝑐 2 − 𝑦 2 −50
(𝑛−8) 2
(𝑥+2) 2
(2𝑡−9) 2
(2𝑥+5)(2𝑥−5)
(7𝑐−12)(7𝑐+12)
2(2𝑦−5)(2𝑦+5)