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 + 2*x* 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:

- Find the greatest common factor which is a whole number (no variables).
- Divide all terms of the polynomial by that factor, and put the result in parentheses. Write the factor outside the parentheses.
- 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.
- Divide each term of the expression in parentheses by the greatest common variable factor, and write the variable factor outside the parentheses.
- Check--distributing the monomial over the new polynomial should yield the original polynomial.

*Example 1*: Factor 4*x*^{2} +16*x*^{3} + 8*x*.

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

*Example 2*: Factor 12*x*^{3}*y* + 3*x*^{4}*y*^{2} -6*x*^{2}*y*^{2}*z*.

- The greatest common whole number factor is 3.
- 12
*x*^{3}*y*+ 3*x*^{4}*y*^{2}-6*x*^{2}*y*^{2}*z*= 3(4*x*^{3}*y*+*x*^{4}*y*^{2}-2*x*^{2}*y*^{2}*z*) - The greatest common variable factor is
*x*^{2}*y*(*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). - 3(4
*x*^{3}*y*+*x*^{4}*y*^{2}-2*x*^{2}*y*^{2}*z*) = 3*x*^{2}*y*(4*x*+*x*^{2}*y*- 2*yz*) - Check: 3
*x*^{2}*y*(4*x*+*x*^{2}*y*- 2*yz*) = 12*x*^{3}*y*+ 3*x*^{4}*y*^{2}-6*x*^{2}*y*^{2}*z*

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