8.1 - Factoring expressions with common factors
This is the easiest form to factor.
An expression is of this form if each term
of the expression contains the same quantity, called the common factor.
This quantity is factored out using the
distributive law.
Here are some examples:
Example: Factor a x − b x
Each term of this expression contains a factor of x,
so this is a common factor. The distributive law allows us to write this
in the factored form:
a x − b x =
x (a − b)
Example: Factor x 3 + 3 x
Again each term contains the common factor of x.
The distributive law allows us to write this as:
x 3 + 3 x =
x (x 2 + 3)
In this example we also had to make use of the
multiplication property of exponents to write
x 3 = x · x 2.
In general, if various terms have various powers of x then the
lowest power of x is the common factor.
Example: Factor 
This expression is more complicated so we will factor it in two steps.
-
The first step is to find the common factor.
The two terms contain the numbers 10 and 15, whose
greatest common factor is 5,
so the common factor must contain 5.
Both terms contain x so the common factor must contain x.
Both terms contain y so the common factor must contain
the lowest power of y, namely y 2.
Thus we can write the factored form of the expression like this:
where the two • quantities that remain inside the brackets
are yet to be determined.
-
The second step is to determine the • quantities. As in the previous
example this is done using the
multiplication property of exponents
for both x and y. The result is that we can write the expression
in factored form like this:
Hint: It is always a good idea to expand or
distribute out the factored form to check that you did the factoring correctly.
Factoring expressions with common factors in denominators
If any term in the expression contains a factor in its denominator then
that factor is considered to be raised to a negative power. As usual when factoring out
a common factor, the lowest power must
be factored out, but now a negative power counts as lower than a positive or zero
power. After the expression has been factored there should be no denominators
remaining inside the brackets. The result should be the same as the result of
adding fractions.
Example: Factor 
Here are the steps:
- Rewrite factors in denominators using
negative exponents.
This shows that the lowest power of x is −4.
- Factor out x to the lowest power. The • quantities are yet
to be determined.
- Use the properties of exponents
to determine the • quantities.
- Use the negative exponent
property to rewrite the common factor as a fraction.
- Optional. Simplify by moving the
bracketted quantity into the numerator. Then we see that the result is
exactly the same as adding fractions
by finding a common denominator!
Example: Factor 
This example is similar to the previous example so study that example first.
The coefficients 1/6 and 1/9
make this example slightly more complicated. Here are the steps:
- Find the common factor. As in the previous example the common factor contains
x 4 in the denominator but what is new is that is also
contains 18 in the denominator. This comes from looking at the
coefficients alone and noticing that to
add them we have to find their lowest common denominator, which is 18:
The general rule is that the denominator of the common factor must contain the
least common multiple or LCM of the denominators
of the coefficients. The LCM of 6 and 9 is 18.
- Determine the • quantities inside the brackets.
- Optional. Move the bracketted quantity into the numerator.
Example: Factor 
This example adds another complication to the previous two examples
(namely the quantities shown below in red)
so study those examples first. Here are the steps:
- Find the common factor. As in the previous example it contains
x 4 and 18 in the denominator but what is new is the
factor shown in red.
- Determine the • quantities inside the brackets. The red and blue
quantities match with the red and blue quantities in the original expression.
- Simplify.
Example: Factor 
This example is probably more easily handled by using the methods of
adding fractions
than by factoring out a common factor. The result is the same.
Nevertheless here are the steps:
- The first term contains a in the denominator. Consider this to the
factor a−1. Even though the other two terms don't
contain a we can consider them to contain the factor
a 0. (Recall that
anything raised to the power 0 equals 1.) Similarly the second term
contains b in the denominator so consider all terms to contain b
raised to some power.
- We see that −1 is the lowest power of a occurring
in all three terms and −1 is the lowest power of b occurring
in all three terms so the common factor is
a−1 b−1.
Factor out the common factor and use the properties of exponents to
determine what quantities remain inside the brackets.
- Use the negative exponent property to rewrite the common factor as a fraction.
- Simplify by moving the bracketted quantity into the numerator.