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CHAPTER 8.4

Learn how to factor special cases such as difference of squares and perfect square trinomials, and use a check list to ensure polynomials are factored correctly.

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CHAPTER 8.4

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  1. CHAPTER 8.4 Special Case Factors!

  2. Recognizing the binomial product Difference of Squares • Recall special case F.O.I.L. where middle terms subtracts out. • e.g. (a+b)(a-b)= a2 - b2 • When factoring Difference of Squares both terms must be perfect squares, the sign must be negative, and when you check by FOIL the middle term subtracts out.

  3. Factor A. x2 – 16B. x2 – 10 C. z6 - 25 • A. (x)2- (4)2 Both terms are perfect squares and sign is ( - ). = (x+4)(x-4) • B. (x)2-(?)2 x is perfect square, but10 is not, therefore NF over Integers. • C. (z3)2-(5)2 Both terms are perfect squares and sign is ( - ). = (z3+5)(z3-5)

  4. Factor A. 25x2 – b2 B. 6x2 – 1 C. n8 - 36 • A. (5x)2- (b)2 Both terms are perfect squares and sign is ( - ). = (5x+b)(5x-b) • B. (?)2-(1)2 6x is not a perfect square, but1 is, therefore NF over Integers. • C. (n4)2-(6)2 Both terms are perfect squares and sign is ( - ). = (n4+6)(n4-6)

  5. Factor z4 - 16 • (z2)2- (4)2 Is the difference of squares and factors to (z2+4)(z2-4) always continue to factor. • (z2+4) (z2-4)second binomial factor is also a difference of squares and factors to(z-2)(z+2). (z2+4) Is simplified! • (z2+4)(z+2)(z-2)

  6. Factor n4 - 81 • (n2)2- (9)2 Is the difference of squares and factors to (n2+9)(n2-9) always continue to factor. • (n2+9) (n2-9)second binomial factor is also a difference of squares and factors to(n+3)(n-3). • (n2+9) Is simplified! • (n2+9)(n+3)(n-3)

  7. Factoring Perfect Square Trinomials • Recall patterns of special case FOIL • (a+b)2 and (a-b)2 • a2+2ab+b2 = (a+b)(a+b) = (a+b)2 • a2 - 2ab+b2 = (a -b)(a -b) = (a-b)2

  8. Things to consider; • Can the first and last terms be rewritten as perfect squares? • Does the middle term equal two times the perfect squares of the first and last terms? • If yes, then you can factor as a perfect square trinomial.

  9. Factor 9x2 – 30x +25 • 1st Rewrite first and last term as perfect squares (3x)2 -30x + (5)2 • 2nd Does 2(a)(b)= middle term? 2(3x)(5)= 30x • Then factor as a perfect square trinomial. • (3x-5)(3x-5) = (3x-5) 2

  10. Factor 16x2 +8x +1 • 1st Rewrite first and last term as perfect squares (4x)2 +8x + (1)2 • 2nd Does 2(a)(b)= middle term? 2(4x)(1)= 8x • Then factor as a perfect square trinomial. • (4x+1)(4x+1) = (4x+1) 2

  11. Factor 4x2 +37x +9 • 1st Rewrite first and last term as perfect squares. (2x)2 +37x + (3)2 • 2nd Does 2(a)(b)= middle term? 2(2x)(3)= 12x = 37x • Then factor is not a perfect square trinomial. Factor by trial and error or grouping.

  12. Factor 4x2 +37x +9 by grouping. • 1st (a)(b) = (4)(9) = 36 • 2nd Consider factors of 36 • 1, 36 • 2, 18 • 3, 12 • 4, 9 • 6, 6 • 3rd Rewrite middle terms 4x2+36x + x +9 • 4th Factor by grouping 4x (x+9) +1(x+9) • (x+9)(4x+1) • Check by F.O.I.L. 4x2+37x+9

  13. Factor x2 +14x +36 • 1st Rewrite first and last term as perfect squares (x)2 +14x + (6)2 • 2nd Does 2(a)(b)= middle term? 2(x)(6)= 12x = 14x • Then factor is not a perfect square trinomial. Factor by trial and error or grouping.

  14. Factor x2 +14x +36 by grouping. • 1st (a)(b) = (1)(36) = 36 • 2nd Consider factors of 36 • 1, 36 • 2, 18 • 3, 12 • 4, 9 • 6, 6 • None of the factors sum to 14, therefore does not factor over integers.

  15. Chapter 8.4 Objective 2 • While factoring, one must always ask are the terms in simplest form? Could the elements be factored further? • This objective will answer that question, and supply a check list for factoring polynomials

  16. Factoring Check List • 1. Is there a common factor? If so, factor the common factor out. • 2. Is the polynomial the difference of two squares? If so, factor. • 3. Is the polynomial a perfect square trinomial? If so, factor. • 4. Is the polynomial a product of two binomials? If so, factor. • 5. Does the polynomial have four terms? If so, try factoring by the grouping method.

  17. Factor 3x2 -48 • 1stGCF 3(x2-16) • Continue to factor as difference of squares. • 3(x+4)(x-4) • Factoring complete, check by FOIL.

  18. Factor x3 -3x2 -4x+12 • 1st There is not a GCF, but contains four terms. Try factor by grouping. x2(x3 -3x2 )-4(-4x+12)= • x2 (x-3)-4(x-3)= • (x-3)(x2 -4)= Continue to factor • (x-3)(x+2)(x-2) • Factoring complete.

  19. Factor 4x2y2 + 12xy2 +9y2 • 1st Factor GCF • y2(4x2 +12x+9)Is remaining trinomial a perfect square trinomial? • First and last term can be written as perfect squares. Two times first times the last perfect squares[2(a)(b)] equals middle term. Therefore, factors as. • y2(2x+3)2

  20. NOW YOU TRY! • 1. x4 -256 • (x2 -16)(x+4)(x-4) • 2. 9x2 -81 • 9(x+3)(x-3) • 3. a2b– 3a2 -16b +48 • (b-3)(a+4)(a-4)

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