Algebra 1 Section 10 1 Factoring is the

Algebra 1 Section 10. 1

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

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

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 8 x 2 – 16. 8 x 2 = 2 • 2 • x 16 = 2 • 2 • 2 GCF = 2 • 2 = 8

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

Factoring Common Monomials from a Polynomial 1. Find the GCF of all terms in the polynomial. 2. Divide the GCF out of each term (applying the Distributive Property in reverse).

Factoring Common Monomials from a Polynomial 3. Write the factorization as the GCF times the sum of the quotients. 4. Check your solution by multiplying the two factors.

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

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

Example 4 Factor 36 x 2 y + 9 xy 3 – 27 xy. GCF = 9 xy(4 x) + 9 xy(y 2) – 9 xy(3) 9 xy(4 x + y 2 – 3)

Example 5 Factor 51 x 3 + 26 y. 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)(3 – y)

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

Homework: pp. 406 -407