Factors
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 processwe 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.
 Checkdistributing the monomial over the new polynomial should yield the
original polynomial.
Example 1: Factor
4x
^{2} +16x
^{3} + 8x
.
 The greatest common whole number factor is
4
.

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

4(x
^{2} +4x
^{3} +2x) = 4x(x + 4x
^{2} + 2)
 Check:
4x(x + 4x
^{2} +2) = 4x
^{2} +16x
^{3} + 8x
Thus,
4x
^{2} +16x
^{3} +8x = 4x(x + 4x
^{2} + 2)
.
Example 2: Factor
12x
^{3}
y + 3x
^{4}
y
^{2} 6x
^{2}
y
^{2}
z
.
 The greatest common whole number factor is
3
.

12x
^{3}
y + 3x
^{4}
y
^{2} 6x
^{2}
y
^{2}
z = 3(4x
^{3}
y + x
^{4}
y
^{2} 2x
^{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(4x
^{3}
y + x
^{4}
y
^{2} 2x
^{2}
y
^{2}
z) = 3x
^{2}
y(4x + x
^{2}
y  2yz)
 Check:
3x
^{2}
y(4x + x
^{2}
y  2yz) = 12x
^{3}
y + 3x
^{4}
y
^{2} 6x
^{2}
y
^{2}
z
Thus,
12x
^{3}
y + 3x
^{4}
y
^{2} 6x
^{2}
y
^{2}
z = 3x
^{2}
y(4x + x
^{2}
y  2yz)
.