Algebra 1 Section 10.1

Factoring Factoring is the process of determining individual factors of a product. We will start by finding greatest common factors.

Example 1 108 = 2 • 2 • 3 • 3 • 3 72 = 2 • 2 • 2 • 3 • 3 GCF = 2 • 2 • 3 • 3 = 36 15x = 3 • 5 • x 42x2y = 2 • 3 • 7 • x • x • y GCF = 3 • x = 3x

Definitions The greatest common factor of a polynomial is the product of all factors shared by every term of the polynomial. Two numbers or terms are considered relatively prime if their GCF is 1.

Example 2 Factor 8x2 – 16. 8x2 = 2 • 2 • 2 • x • x 16 = 2 • 2 • 2 • 2 GCF = 2 • 2 • 2 = 8

Example 2 Factor 8x2 – 16. GCF = 2 • 2 • 2 = 8 Factor 8 from each term. 8(x2) – 8(2) 8(x2 – 2)

Factoring Common Monomials from a Polynomial Find the GCF of all terms in the polynomial. Divide the GCF out of each term (applying the Distributive Property in reverse).

Factoring Common Monomials from a Polynomial Write the factorization as the GCF times the sum of the quotients. Check your solution by multiplying the two factors.

Example 3 Factor 42a3 + 14a. 42a3 = 2 • 3 • 7 • a • a • a GCF = 2 • 7 • a = 14a

Example 3 Factor 42a3 + 14a. GCF = 2 • 7 • a = 14a Factor 14a from each term. 14a(3a2) + 14a(1) 14a(3a2 + 1)

Example 4 Factor 36x2y + 9xy3 – 27xy. GCF = 9xy 9xy(4x) + 9xy(y2) – 9xy(3) 9xy(4x + y2 – 3)

Example 5 Factor 51x3 + 26y. GCF = 1 Since the two terms are relatively prime, the polynomial is prime and will not factor.

Factoring A polynomial can contain a common binomial factor. 3(x + y) – y(x + y) (x + y)(3 – y)

Example 6 Factor x(x + 1) + 2(x + 1). (x + 1) is a factor of each term. (x + 1)(x + 2)

Homework: pp. 406-407