Section 6-7

Factoring (Greatest Common Factors)

FINDING THE PRIME FACTORIZATION OF A WHOLE NUMBER

To find the prime factorization of a whole number, first see if it's divisible by two. If it is, write it as a product of two and another number. Then see if the second number is divisible by two. Repeat until you get an odd number.

Then, see if the remaining number is divisible by three. If it is, write it as a product of three and another number. Then see if the remaining number is divisible by three. Repeat until you get a number which isn't.

Repeat this process for each prime number (2, 3, 5, 7, 11, 13, 17, etc.) until you are left with a product of all prime numbers. This is the prime factorization.

First, find out how many "2-factors" there are.

504

= 2 × 252

= 2 × 2 × 126

= 2 × 2 × 2 × 63

Now, find out how many "3-factors" there are.

= 2 × 2 × 2 × 3 × 21

= 2 × 2 × 2 × 3 × 3 × 7

FINDING THE PRIME FACTORIZATION OF A MONOMIAL

To find the prime factorization of a monomial, first find the prime factorization of the coefficient (if it is a whole number). If the coefficient is negative, factor out a –1. Then, write out all the variable factors individually (rather than with exponents.)

First, find the prime factorization of the coefficient.

–44x³yz² = –1 · 2 · 2 · 11 · x³yz²

Now break up the variable expression.

–44x³yz² = –1 · 2 · 2 · 11 · x · x · x · y · z · z

FINDING THE GREATEST COMMON FACTOR OF TWO MONOMIALS

To find the GCF of two monomials, first factor them completely, and then find the product of all the common factors.

First, find the prime factorization each monomial.

–27p²qr⁵ = –1 · 3 · 3 · 3 · p · p · q · r · r · r · r · r

15p³r³ = 3 · 5 · p · p · p · r · r · r

Common factors are shown in red. Their product is:

3 · p · p · r · r · r

So, the GCF is 3p²r³.