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.
Removing Common Factors
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:
- 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 4x2 +16x3 + 8x.
- The greatest common whole number factor is 4.
- 4x2 +16x3 +8x = 4(x2 +4x3 + 2x)
- The greatest common variable factor is x (x is contained in all the terms, and its lowest exponent is 1).
- 4(x2 +4x3 +2x) = 4x(x + 4x2 + 2)
- Check: 4x(x + 4x2 +2) = 4x2 +16x3 + 8x
Example 2: Factor 12x3y + 3x4y2 -6x2y2z.
- The greatest common whole number factor is 3.
- 12x3y + 3x4y2 -6x2y2z = 3(4x3y + x4y2 -2x2y2z)
- 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).
- 3(4x3y + x4y2 -2x2y2z) = 3x2y(4x + x2y - 2yz)
- Check: 3x2y(4x + x2y - 2yz) = 12x3y + 3x4y2 -6x2y2z