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Published byGyles Fleming Modified over 6 years ago

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**Perfect Square Trinomials and Difference of Perfect Squares**

Factoring Perfect Square Trinomials and Difference of Perfect Squares

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**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

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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

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

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**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)

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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)

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

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**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

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**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)

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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

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**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

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**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)

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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)

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