Factoring Perfect Square

Trinomials and Difference of Perfect Squares

STANDARD 4.0 Factor with special patterns

Factor the expression.a. x2 – 49

= (x + 7)(x – 7)Difference of two squares

b. d 2 + 12d + 36

= (d + 6)2

Perfect square trinomial

c. z2 – 26z + 169

= (z – 13)2

Perfect square trinomial

= x2 – 72

= d 2 + 2(d)(6) + 62

= z2 – 2(z) (13) + 132

GUIDED PRACTICE for Example 2

4. x2 – 9

(x – 3)(x + 3)

5. q2 – 100

(q – 10)(q + 10)

6. y2 + 16y + 64

(y + 8)2

Factor the expression.

ANSWER

ANSWER

ANSWER

GUIDED PRACTICE for Example 2

7. w2 – 18w + 81

(w – 9)2

STANDARD 4.0 Factor out monomials first

Factor the expression.

a. 5x2 – 45

= 5(x + 3)(x – 3)

b. 6q2 – 14q + 8

= 2(3q – 4)(q – 1)

c. –5z2 + 20z

d. 12p2 – 21p + 3

= 5(x2 – 9)

= 2(3q2 – 7q + 4)

= –5z(z – 4)

= 3(4p2 – 7p + 1)

GUIDED PRACTICEGUIDED PRACTICE for Example 4

Factor the expression.

13. 3s2 – 24

14. 8t2 + 38t – 10

2(4t – 1) (t + 5)

3(s2 – 8)

15. 6x2 + 24x + 15

3(2x2 + 8x + 5)

16. 12x2 – 28x – 24

4(3x + 2)(x – 3)

17. –16n2 + 12n

–4n(4n – 3)ANSWER

ANSWER

ANSWER

ANSWER

ANSWER

GUIDED PRACTICEGUIDED PRACTICE for Example 4

18. 6z2 + 33z + 36

3(2z + 3)(z + 4)

ANSWER

STANDARD 4.0 Factor by grouping

Factor the polynomial x3 – 3x2 – 16x + 48 completely.

x3 – 3x2 – 16x + 48 Factor by grouping.

= (x2 – 16)(x – 3) Distributive property

= (x + 4)(x – 4)(x – 3) Difference of two squares

= x2(x – 3) – 16(x – 3)

Factor polynomials in quadratic form

Factor completely: (a) 16x4 – 81 and (b) 2p8 + 10p5 + 12p2.

a. 16x4 – 81 Write as differenceof two squares.

= (4x2 + 9)(4x2 – 9) Difference of two squares

= (4x2 + 9)(2x + 3)(2x – 3) Difference of two squares

b. 2p8 + 10p5 + 12p2 Factor common monomial.

= 2p2(p3 + 3)(p3 + 2) Factor trinomial in quadratic form.

= (4x2)2 – 92

= 2p2(p6 + 5p3 + 6)

GUIDED PRACTICE for Examples 3 and 4

Factor the polynomial completely.

5. x3 + 7x2 – 9x – 63

(x + 3)(x – 3)(x + 7)

6. 16g4 – 625

(4g2 + 25)(2g + 5)(2g – 5)

7. 4t6 – 20t4 + 24t2

4t2(t2 – 3)(t2 – 2 )ANSWER

ANSWER

ANSWER

EXAMPLE 1 Find a common monomial factor

Factor the polynomial completely.

a. x3 + 2x2 – 15x Factor common monomial.

= x(x + 5)(x – 3) Factor trinomial.

b. 2y5 – 18y3 Factor common monomial.

= 2y3(y + 3)(y – 3) Difference of two squares

c. 4z4 – 16z3 + 16z2 Factor common monomial.

= 4z2(z – 2)2 Perfect square trinomial

= x(x2 + 2x – 15)

= 2y3(y2 – 9)

= 4z2(z2 – 4z + 4)

EXAMPLE 2 Factor the sum or difference of two cubes

Factor the polynomial completely.

a. x3 + 64

= (x + 4)(x2 – 4x + 16)

Sum of two cubes

b. 16z5 – 250z2 Factor common monomial.

= 2z2 (2z)3 – 53 Difference of two cubes

= 2z2(2z – 5)(4z2 + 10z + 25)

= x3 + 43

= 2z2(8z3 – 125)

GUIDED PRACTICE for Examples 1 and 2

Factor the polynomial completely.

1. x3 – 7x2 + 10x

x( x – 5 )( x – 2 )2. 3y5 – 75y3

3y3(y – 5)(y + 5 )

3. 16b5 + 686b2

2b2(2b + 7)(4b2 –14b + 49)

4. w3 – 27

(w – 3)(w2 + 3w + 9)ANSWER

ANSWER

ANSWER

ANSWER