Multiplication of polynomials relies on the distributive property. The reverse process, where a polynomial is written as a product of other polynomials, is called factoring. For example, one way to factor the number 18 is to write it as the product both 9 and 2 are factors of 18. Usually, only integers are used as factors of integers. The number 18 can also be written with three integer factors as

The Greatest Common Factor

To factor the algebraic expression 15m + 45, first note that both 15 m and 45 are divisible by 15; and By the distributive property,

Both 15 and m+3 are factors of 15m + 45. Since 15 divides into both terms of 15m + 45 (and is the largest number that will do so), 15 is the greatest common factor for the polynomial 15m + 45. The process of writing 15m + 45 as15(m+3) is often called factoring out the greatest common factor.

EXAMPLE

Factoring

Factor out the greatest common factor.

(a) 12p - 18q

Solution

Both 12 p and 18 q are divisible by 6. Therefore,

12p - 18q = 6·2p - 6·3q = 6 (2p -3q)

(b) 8x- 9x + 15x

Solution

Each of these terms is divisible by x .

8x- 9x + 15x = (8x)x- (9x)x + (15)x

= x(8x- 9x + 15) or (8x- 9x + 15)x

One can always check factorization by finding the product of the factors and comparing it to the original expression.

CAUTION

When factoring out the greatest common factor in an expression like 2x + x be careful to remember the 1 in the second term.

2x + x = 2x + 1x = x(2x + 1) not x(2x)