Factoring Simple Polynomials

Definitions Factors of a number are the numbers that divide the original number evenly. Writing a number as a product of factors is called a factorization of the number. The prime factorization of a number is the factorization of that number written as a product of prime numbers. Common factors are factors that two or more numbers have in common. The Greatest Common Factor (GCF) is the largest common factor.

Ex: Find the GCF(24, 40). Prime factor each number: 24 2 12 2 6 2 3  24 = 2*2*2*3 = 23*3  GCF(24,40) = 23 = 8 40 2 20 2 10 2 5  40 = 2*2*2*5 = 23*5

The Greatest Common Factor of terms of a polynomial is the largest factor that the original terms share Ex: What is the GCF(7x2, 3x) 7x2 = 7 * x * x 3x = 3 * x The terms share a factor of x  GCF(7x2, 3x) = x

Greatest Common Factors aka GCF’s Find the GCF for each set of following numbers. Find means tell what the terms have in common. Hint: list the factors and find the greatest match. 2, 6 -25, -40 6, 18 16, 32 3, 8 2 -5 6 16 1 No common factors? GCF =1

Greatest Common Factors aka GCF’s Find the GCF for each set of following numbers. Hint: list the factors and find the greatest match. x, x2 x2, x3 xy, x2y 2x3, 8x2 3x3, 6x2 4x2, 5y3 x x2 xy 2x2 3x2 1 No common factors? GCF =1

Greatest Common Factors aka GCF’s Factor out the GCF for each polynomial: Factor out means you need the GCF times the remaining parts. How can you check? a) 2x + 4y 5a – 5b 18x – 6y 2m + 6mn 5x2y – 10xy 2(x + 2y) 5(a – b) 6(3x – y) 2m(1 + 3n) 5xy(x - 2)

FACTORING by GCF Take out the GCF EX: 15xy2 – 10x3y + 25xy3 How: Find what is in common in each term and put in front. See what is left over. Check answer by distributing out. Solution: 5xy( ) 3y – 2x2 + 5y2

FACTORING Take out the GCF EX: 2x4 – 8x3 + 4x2 – 6x How: Find what is in common in each term and put in front. See what is left over. Check answer by distributing out. Solution: 2x (x3 – 4x2 + 2x – 3)

Ex: Factor 6a5 – 3a3 – 2a2 Recall: GCF(6a5,3a3,2a2)= a2 1 6a5 – 3a3 – 2a2 = a2( - - ) 6a3 3a 2 a2 a2 a2  6a5 – 3a3 – 2a2 = a2(6a3 – 3a – 2)