Factoring

Common Monomial Factoring always check for first (reverse of Distributive Property; factor out the common stuff) 6x – 9 = 2·3·x - 3·3 = 3(2x – 3) 5x2 + 8x = 5·x·x + 2·2·2·x = x(5x+8) 10x3–15x2=2·5·x·x·x-3·5·x·x=5x2(2x-3) x2 + 3x – 4 = x·x + 3·x - 2·2 = x2 + 3x – 4 (nothing common)

Factor by Grouping (4 terms) Group first two terms; make sure third term is addition; group last two terms Common Monomial Factor both parentheses (inside stuff must be same in both parentheses) Answer: (Outside stuff)·(Inside stuff) 5x2 – 3x – 10x + 6 = (5x2 – 3x) + (–10x + 6) = x(5x-3) – 2(5x – 3) = (x – 2)(5x – 3)

unFOILing: Guess & Check method (3 terms: ax2 + bx + c) List factor pairs of a; these are the possible coefficients of x in the two parentheses. List factor pairs of c; these are the possible constant terms in the two parentheses. Guess by combining the factor pairs of both a & c then compare the sum of the Outer and Inner multiplications to b. If the check works you have your answer; if not guess again. a=2 b=7 c=-15 Factor pairs of a c 2 1 -15 -5 -3 15 -1 2x2 + 7x – 15 Guess #1: (x -1)(2x + 15) = 2x2 + 15x – 2x – 15 = 2x2 + 13x – 15 error Guess #2: (x + 5)(2x – 3) = 2x2 -3x + 10x – 15 = 2x2 + 7x – 15 correct Therefore (x + 5)(2x – 3) is your answer.

unFOILing without shortcut method (3 terms: ax2 + bx + c) Find two numbers r & s, so that r + s = b and r ·s = a · c a=2 b=7 c=-15 r+s = 7 r·s = 2(-15) = -30 -1·-30=-30 1+-30=-29 2 ·-15=-30 2+-15=-13 3 ·-10=-30 3+-10=-7 5·-6=-24 5+-6=-1 2x2 + 7x – 15 2x2 - 3x + 10x – 15 (2x2 - 3x) + (10x – 15) x(2x - 3) + 5(2x – 3) (x + 5)(2x – 3)

unFOILing with shortcut (3 terms: x2 + bx + c; a=1) Find two numbers, r & s, so that r + s = b and r · s = c Answer: (x + r)(x + s) x2 + 5x – 24 = (x+8)(x-3) a=1 b=5 c=-24 r+s = 5 r·s = -24 1·-24=-24 1+-24=-23 2 ·-12=-24 2+-12=-10 3 ·-8=-24 3+-8=-5 4 ·-6=-24 4+-6=-2 6 ·-4=-24 6+-4=2 8 ·-3=-24 8+-3=5 12 ·-2=-24 12+-2=10 24·-1=-24 24+-1=23

Difference of two squares Two terms: a2-b2 x2 x 1 4 2 9 3 16 25 5 36 6 49 7 64 8 81 100 10 121 11 144 12 Find square roots of both terms Answer: (a + b)(a – b) 25x2 - 49 =(5x)2 – (7)2 =(5x + 7)(5x – 7)

Flowchart Common Monomial Number of terms 2 3 4 Difference of 2 3 4 Difference of Two Squares Factor by Grouping Does a = 1? yes no r & s method with shortcut Do you know square root of first and last terms? yes no r & s method without shortcut Perfect Square Trinomial Does 2ab part work? yes no