Factoring Advanced Algebra Chapter 5

Factoring & Roots  Factors  numbers, variables, monomials, or polynomials multiplied to obtain a product.  Prime Number  a whole number greater than 1 whose only factors are 1 and itself.  Polynomials that cannot be factored are called prime.

Factoring & Roots Table of Most Common Factoring Techniques Any Number of Terms Greatest Common Factor (GCF)a 3 b 2 + 2a 2 b – 4ab 2 = ab(a 2 b + 2a – 4b) Two Terms Difference of Two Squares Sum of Two Cubes Difference of Two Cubes a 2 – b 2 = (a + b)(a – b) a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) Three Terms Perfect Square Trinomials a 2 + 2ab + b 2 = (a + b) 2 a 2 – 2ab + b 2 = (a – b) 2 General Trinomialsacx 2 + (ad + bc)x + bd = (ax + b)(cx + d) Four or More Terms Grouping ra + rb + sa + sb = r(a + b) + s(a + b) = (r + s)(a + b)

Examples of Factoring  Greatest Common Factor (GCF) 14a 3 b 3 c – 21a 2 b 4 c + 7a 2 b 3 c The GCF is 7a 2 b 3 c After Factoring the result is 7a 2 b 3 c(2a – 3b + 1)

In-Class Practice (Factor)

Examples of Factoring  Four or More Terms (Grouping) 7ax + 2bx – 7ay 2 – 2by 2 Grouping result in (7ax + 2bx) – (7ay 2 + 2by 2 ) Factoring in each group results in x (7a + 2b) – y 2 (7a + 2b) Final factoring has a result of (7a + 2b)(x – y 2 )

In-Class Practice (Grouping)

Examples of Factoring 3x x + 20  Let’s analyze. The product of the coefficient of the first term and constant term is 3 ∙ 20 or 60.  If we were to split the middle x term the sum of the two coefficients must be equal to 17.  Best guess is that the two coefficients of the x terms should be 5 and 12 since (5)(12) = 60 and = 17  By separating the x term and regrouping we can rewrite the expression as follows: 3x x + 5x + 20 =3x(x+4)+5(x+4) =(3x+5)(x+4)

In-Class Practice (Trinomial) Note: it works to factor the 2 out at the end as well.