Suppose that

x = a*b

and suppose that

y = a*c

We have

With numbers, that might look like

10 = 2*5

14 = 2*7

2 is a common factor of 10 and 14.

Applying this idea to something a little more complicated, we might have

y = x^2 + 4x - x - 4

We can factor this as

y = x(x+4) - 1(x+4)

The factor (x+4) is common to the two terms we have created. It can be factored out, so we get

y = (x-1)(x+4)

We can take this another step further. Consider the expression

(x^2 + 3x - 4)/(x^2 - 16)

We can factor the numerator and the denominator.

((x-1)(x+4))/((x-4)(x+4))

We can see that (x+4) is a factor that is common to both the numerator and denominator. We can use that fact to simplify the expression. The ratio of identical factors is always 1.

((x-1)(x+4))/((x-4)(x+4)) = ((x-1)/(x-4))*((x+4)/(x+4)) = ((x-1)/(x-4))*1 = (x-1)/(x-4)

##### We used the common factor to simplify the process of complete factoring, and we used the common factors to simplify an expression with common factors in numerator and denominator. There are, no doubt, other uses.

x = a*b

and suppose that

y = a*c

We have

**a**as a factor of x, and**a**is also a factor of y. We say that**a**is a common factor of x and y. Those two products have that factor (a) in common.With numbers, that might look like

10 = 2*5

14 = 2*7

2 is a common factor of 10 and 14.

Applying this idea to something a little more complicated, we might have

y = x^2 + 4x - x - 4

We can factor this as

y = x(x+4) - 1(x+4)

The factor (x+4) is common to the two terms we have created. It can be factored out, so we get

y = (x-1)(x+4)

We can take this another step further. Consider the expression

(x^2 + 3x - 4)/(x^2 - 16)

We can factor the numerator and the denominator.

((x-1)(x+4))/((x-4)(x+4))

We can see that (x+4) is a factor that is common to both the numerator and denominator. We can use that fact to simplify the expression. The ratio of identical factors is always 1.

((x-1)(x+4))/((x-4)(x+4)) = ((x-1)/(x-4))*((x+4)/(x+4)) = ((x-1)/(x-4))*1 = (x-1)/(x-4)