Perfect Square Trinomials and Difference of Perfect Squares

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Presentation transcript:

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

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 2 + 12d + 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

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

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

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)

GUIDED PRACTICE GUIDED PRACTICE for Example 4 Factor the expression. 13. 3s2 – 24 ANSWER 3(s2 – 8) 14. 8t2 + 38t – 10 ANSWER 2(4t – 1) (t + 5) 15. 6x2 + 24x + 15 ANSWER 3(2x2 + 8x + 5) 16. 12x2 – 28x – 24 ANSWER 4(3x + 2)(x – 3) 17. –16n2 + 12n ANSWER –4n(4n – 3)

GUIDED PRACTICE GUIDED PRACTICE for Example 4 18. 6z2 + 33z + 36 ANSWER 3(2z + 3)(z + 4)

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

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

GUIDED PRACTICE for Examples 3 and 4 Factor the polynomial completely. 5. x3 + 7x2 – 9x – 63 ANSWER (x + 3)(x – 3)(x + 7) 6. 16g4 – 625 (4g2 + 25)(2g + 5)(2g – 5) ANSWER 7. 4t6 – 20t4 + 24t2 4t2(t2 – 3)(t2 – 2 ) ANSWER

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

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. 16z5 – 250z2 = 2z2(8z3 – 125) Factor common monomial. = 2z2 (2z)3 – 53 Difference of two cubes = 2z2(2z – 5)(4z2 + 10z + 25)

GUIDED PRACTICE for Examples 1 and 2 Factor the polynomial completely. 1. x3 – 7x2 + 10x ANSWER x( x – 5 )( x – 2 ) 2. 3y5 – 75y3 ANSWER 3y3(y – 5)(y + 5 ) 3. 16b5 + 686b2 ANSWER 2b2(2b + 7)(4b2 –14b + 49) 4. w3 – 27 ANSWER (w – 3)(w2 + 3w + 9)