1. Perfect Square Trinomials and Difference of Perfect Squares
Factoring
Perfect Square Trinomials and Difference of Perfect Squares

2. Factor with special patterns
STANDARD 4.0
Factor with special patterns
Factor the expression.
a. x2 – 49
= x2 – 72
Difference of two squares
= (x + 7)(x – 7)
b. d d + 36
= d 2 + 2(d)(6) + 62
Perfect square trinomial
= (d + 6)2
c. z2 – 26z + 169
= z2 – 2(z) (13) + 132
Perfect square trinomial
= (z – 13)2

3. GUIDED PRACTICE
for Example 2
Factor the expression.
4. x2 – 9
ANSWER
(x – 3)(x + 3)
5. q2 – 100
ANSWER
(q – 10)(q + 10)
6. y2 + 16y + 64
ANSWER
(y + 8)2

4. GUIDED PRACTICE
for Example 2
7. w2 – 18w + 81
(w – 9)2

5. Factor out monomials first
STANDARD 4.0
Factor out monomials first
Factor the expression.
a. 5x2 – 45
= 5(x2 – 9)
= 5(x + 3)(x – 3)
b. 6q2 – 14q + 8
= 2(3q2 – 7q + 4)
= 2(3q – 4)(q – 1)
c. –5z2 + 20z
= –5z(z – 4)
d. 12p2 – 21p + 3
= 3(4p2 – 7p + 1)

6. GUIDED PRACTICE
GUIDED PRACTICE
for Example 4
Factor the expression.
s2 – 24
ANSWER
3(s2 – 8)
t2 + 38t – 10
ANSWER
2(4t – 1) (t + 5)
x2 + 24x + 15
ANSWER
3(2x2 + 8x + 5)
x2 – 28x – 24
ANSWER
4(3x + 2)(x – 3)
–16n2 + 12n
ANSWER
–4n(4n – 3)

7. GUIDED PRACTICE
GUIDED PRACTICE
for Example 4
z2 + 33z + 36
ANSWER
3(2z + 3)(z + 4)

8. Factor the polynomial x3 – 3x2 – 16x + 48 completely.
STANDARD 4.0
Factor by grouping
Factor the polynomial x3 – 3x2 – 16x + 48 completely.
x3 – 3x2 – 16x + 48
= x2(x – 3) – 16(x – 3)
Factor by grouping.
= (x2 – 16)(x – 3)
Distributive property
= (x + 4)(x – 4)(x – 3)
Difference of two
squares

9. Factor polynomials in quadratic form
Factor completely: (a) 16x4 – 81 and (b) 2p8 + 10p5 + 12p2.
a x4 – 81
= (4x2)2 – 92
Write as difference
of two squares.
= (4x2 + 9)(4x2 – 9)
Difference of two
squares
= (4x2 + 9)(2x + 3)(2x – 3)
Difference of two
squares
Factor common
monomial.
b. 2p8 + 10p5 + 12p2
= 2p2(p6 + 5p3 + 6)
Factor trinomial in
quadratic form.
= 2p2(p3 + 3)(p3 + 2)

10. GUIDED PRACTICE
for Examples 3 and 4
Factor the polynomial completely.
x3 + 7x2 – 9x – 63
ANSWER
(x + 3)(x – 3)(x + 7)
g4 – 625
(4g2 + 25)(2g + 5)(2g – 5)
ANSWER
t6 – 20t4 + 24t2
4t2(t2 – 3)(t2 – 2 )
ANSWER

11. Find a common monomial factor
EXAMPLE 1
Find a common monomial factor
Factor the polynomial completely.
a. x3 + 2x2 – 15x
= x(x2 + 2x – 15)
Factor common
monomial.
= x(x + 5)(x – 3)
Factor trinomial.
b. 2y5 – 18y3
= 2y3(y2 – 9)
Factor common
monomial.
= 2y3(y + 3)(y – 3)
Difference of two
squares
c. 4z4 – 16z3 + 16z2
= 4z2(z2 – 4z + 4)
Factor common
monomial.
= 4z2(z – 2)2
Perfect square trinomial

12. Factor the sum or difference of two cubes
EXAMPLE 2
Factor the sum or difference of two cubes
Factor the polynomial completely.
a. x3 + 64
= x3 + 43
Sum of two cubes
= (x + 4)(x2 – 4x + 16)
b z5 – 250z2
= 2z2(8z3 – 125)
Factor common monomial.
= 2z2 (2z)3 – 53
Difference of two cubes
= 2z2(2z – 5)(4z2 + 10z + 25)

13. GUIDED PRACTICE
for Examples 1 and 2
Factor the polynomial completely.
x3 – 7x2 + 10x
ANSWER
x( x – 5 )( x – 2 )
y5 – 75y3
ANSWER
3y3(y – 5)(y + 5 )
b b2
ANSWER
2b2(2b + 7)(4b2 –14b + 49)
4. w3 – 27
ANSWER
(w – 3)(w2 + 3w + 9)