Factoring Perfect Square Trinomials and Difference of Perfect Squares
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Transcript of Factoring Perfect Square Trinomials and Difference of Perfect Squares
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.
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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
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GUIDED PRACTICEGUIDED PRACTICE for Example 4
18. 6z2 + 33z + 36
3(2z + 3)(z + 4)
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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
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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
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