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Factors

In general, a factor of a number x is a number that can be multiplied by another number to get x .   But, the definition changes a little according to what kind of math you're doing.

For whole numbers a and n :

We say a is a factor of n if a b = n for some whole number b .

For example, 3 is a factor of 21 , since 3 7 = 21 .

But 4 is not a factor of 21 , since there is no whole number b for which 4 b = 21 .

For polynomials p and r :

We say p is a factor of r if p q = r for some polynomial r .

For example, x + 1 is a factor of x 2 2 x 3 , since

( x + 1 ) ( x 3 ) = x 2 2 x 3 .

But x + 2 is not a factor of x 2 2 x 3 , since there is no polynomial q for which ( x + 2 ) ( q ) = x 2 2 x 3 .

Common Factors

If two numbers (or polynomials) have a factor in common, then it is called a common factor.

For instance, the numbers 15 and 33 have 3 as a common factor.

The polynomials

4 x + 4  and   x 2 2 x 3

have x + 1 as a common factor.

4 x + 4 = 4 ( x + 1 )

x 2 2 x 3 = ( x + 1 ) ( x 3 )