How to Factor!

How to Factor! How to Factor •*Always work in this order 1) GCF: Greatest common factor • - it may be a constant, a variable, of a combination of both (3, X, 4X) • Ex: x+7x = x(x+7) • 2x-8x = 2x(x-4) • 3x-6x+3 = 3(x-2x+1) 2) DOS: Check for a Difference of Squares • - Is the first a perfect square? • - Is the second a perfect square? • - Is there a minus between the two terms? • - Are there only two terms? • If so: factor as • (root of 1st + root of 2nd)(root 1 – root 2) • Ex: 4x 2- 361 = • (2x+19)(2x-19) 3) PTS: Perfect Trinomial Square • - Is the first a perfect square? • - Is the last a perfect square? • - is the middle 2* root 1st*root last? • If so, it factors as • (root of 1st ± root of last)2 • Use the middle sign PTS cont. • Ex: x 2 -10x + 25 • (x-5)2 • x 2 + 20x + 64 • not a PTS 2*x*5 = 10x yes 2*x*8 = 16x no 4) 2S, 2D, or S&D: (1st coefficient is a 1) • If: • all terms are positive It factors: 2 sums - only the middle term is negative 2 differences - the last term is negative, regardless of the middle 1 sum & 1 difference 4) 2S, 2D, or S&D: (1st coefficient is a 1) cont… • Ex: x 2 - 6x - 16 • • • • • Factors as: • (x+2)(x-8) S&D -16: -1, 16 -2, 8 -4, 4 2, -8 (2-8=-6 the middle term) Bustin’ da B!! • This method is used for trinomials in which the first and last term have coefficients other than 1, and it’s not a PTS. Bustin’ da B!! Cont… • Ex: 2x 2 - 5x – 12 • • • -2, 12 • -3, 8 • -4, 6 • Bust da B term (-5x) into: • 3x + -8x • • in the original problem • 1, -24 2, -12 3, -8 (3 -8 = -5) Multiply 2*-12 = -24 (first coeff x last coeff) write it’s factors -24: -1, 24 Bustin’ da B!! Cont… • • • • • 2x 2 - 5x - 12 2x 2 + 3x + -8x -12 (2x 2 + 3x) + (-8x-12) x(2x+3) - 4(2x+3) (2x+3) (x-4) Original Bust B. Group. Factor GCF. Write what they have in common, then write what’s left. 6) Grouping • Used when there are 4 or more terms. • Ex: 2ab + 14a + b + 7 • (2ab +14a) + (b + 7) • 2a(b+7) + 1(b+7) • (b+7)(2a+1) Write what they have in common, then write what’s left. Grouping cont…. • Ex: xy + 3x – y2- 3y • (xy + 3x) + (-y2-3y)------always use a plus sign between • parenthesis • x(y+3) - y(y+3) GCF each Parenthesis • (y+3)(x-y) Write what they have in common, then write what’s left. 7) Sum/Difference of 2 cubes • a 3+ b 3 = (a + b)(a 2- ab + b 2) • Ex: 64x 3 + 27 = (4x + 3)(16x 2 -12x + 9) 3 3 • 27 = 3 64x3 = 4x • a 3 – b 3 = (a-b)(a 2 + ab + b 2) • • Ex: 8x 3-125 = (2x – 5)(4x 2+ 10x + 25) 3 3 125 = 5 8x3 = 2x