A factor is a number that evenly divides the given number. A factor need not be a constant. In fact, any integer, variable, or polynomial that can be multiplied by an integer, a variable, or a polynomial to produce the given expression is a factor of the given expression.

We've seen how to distribute a quantity over a polynomial and write the result as a polynomial. We can actually reverse this process--we can "remove" a common factor from a polynomial and write the result as a quantity times a polynomial. For example, 12 + 2x can be written as 2(6 + x).

The first step to removing a common factor is finding a common factor. A common factor is a factor of all the terms in an expression (i.e., a factor that they all have in common). A common factor can be an integer, a variable, or a combination of integers and variables.

To remove a common factor and rewrite a polynomial as the product of a monomial and another polynomial:

1. Find the greatest common factor which is a whole number (no variables).
2. Divide all terms of the polynomial by that factor, and put the result in parentheses. Write the factor outside the parentheses.
3. Find the greatest common factor which is a variable or a product of several variables. That is, find the variables contained in every term, and write them with their lowest exponent.
4. Divide each term of the expression in parentheses by the greatest common variable factor, and write the variable factor outside the parentheses.
5. Check--distributing the monomial over the new polynomial should yield the original polynomial.

Example 1: Factor 4x² +16x³ + 8x.

1. The greatest common whole number factor is 4.
2. 4x² +16x³ +8x = 4(x² +4x³ + 2x)
3. The greatest common variable factor is x (x is contained in all the terms, and its lowest exponent is 1).
4. 4(x² +4x³ +2x) = 4x(x + 4x² + 2)
5. Check: 4x(x + 4x² +2) = 4x² +16x³ + 8x

Thus, 4x² +16x³ +8x = 4x(x + 4x² + 2).

Example 2: Factor 12x³y + 3x⁴y² -6x²y²z.

1. The greatest common whole number factor is 3.
2. 12x³y + 3x⁴y² -6x²y²z = 3(4x³y + x⁴y² -2x²y²z)
3. The greatest common variable factor is x²y (x is contained in all the terms, and its lowest exponent is 2; y is contained in all the terms, and its lowest exponent is 1; z is not contained in all the terms).
4. 3(4x³y + x⁴y² -2x²y²z) = 3x²y(4x + x²y - 2yz)
5. Check: 3x²y(4x + x²y - 2yz) = 12x³y + 3x⁴y² -6x²y²z

Thus, 12x³y + 3x⁴y² -6x²y²z = 3x²y(4x + x²y - 2yz).