8-7 Factoring Special Cases
Hubarth
Algebra
Perfect Squares Trinomials
For every real number a and b:
𝑎2 + 2𝑎𝑏 + 𝑏2 = 𝑎 + 𝑏 𝑎 + 𝑏 = (𝑎 + 𝑏)2
𝑎2 − 2𝑎𝑏 + 𝑏2 = 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.
Ex 1 Factoring Perfect Squares
Factor 𝑥 2 + 8𝑥 + 16
𝑥 2 is a perfect square
16 is a perfect square
Our factors are
(x + 4)(x + 4) = (𝑥 + 4)2
the square root of 𝑥 2 is x
the square root of 16 is 4
2(4)(x) = 8x
our middle term
Ex 2 Factoring Perfect Squares
Factor 9𝑔2 − 12𝑔 + 4
9𝑔2 is perfect square
4 is a perfect square
9𝑔2 = 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
Difference of Squares
For every real number a and b:
𝑎2 − 𝑏2 = (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
Ex 3 Factor the Difference of Squares
Factor 𝑥 2 − 16
𝑥 − 4 (𝑥 + 4)
Ex 4 Factor the Difference of Squares
Factor 4𝑥 2 − 121
(2𝑥 + 11)(2𝑥 − 11)
Ex 5 Factoring Out a Common Factor
Factor 10𝑥 2 − 40
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)
Practice
Factor each expression
1. 𝑛2 − 16𝑛 + 64
(𝑛 − 8)2
2. 𝑥 2 + 4𝑥 + 4
3. 4𝑡 2 − 36𝑡 + 81
(𝑥 + 2)2
(2𝑡 − 9)2
4. 4𝑥 2 − 25
5. 49𝑐 2 − 144
(2𝑥 + 5)(2𝑥 − 5)
(7𝑐 − 12)(7𝑐 + 12)
6. 8𝑦 2 − 50
2(2𝑦 − 5)(2𝑦 + 5)