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 1: 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 2: 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 3: 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
Example 2 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.
Common factors in the denominator
If an expression is made up of fractions and if every fraction’s denominator contains the same quantity, then
that quantity
can also be factored out with the distributive law.
Example 4: Factor
Here are the steps:
 One denominator is x^{2} and the other denominator is x^{4} and their common factor is x^{2}. We put this in the factor’s denominator. The • quantities are yet to be determined.
 Use the properties of exponents
to determine the • quantities.
Example 5: Factor
Here are the steps:
 Find the common factor in the denominators. It is 3x^{ 2} and we put this in the factor’s denominator. The • quantities are yet to be determined.
 Use the distributive law to determine the • quantities.
Example 6: 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 3x^{ 2} in the denominator but what is new is the quantity
5a shown in red. The • quantities are yet to be determined.
 Determine the • quantities inside the brackets. The red and blue
quantities match with the red and blue quantities in the original expression.
‘Uncommon’ Common Factors
Sometimes it is desirable to factor out some quantity from an expression even if not every term contains that quantity. The usual reason is to achieve some desired form. Here are some examples:
Example 7: Suppose that we want to apply the
completingthesquare process (this is
covered in the next chapter
) to the quadratic
2 x^{ 2} + 9 x − 20.
The problem is that this process requires that we start with a quadratic of the form
x^{ 2} + b x + c.
(That is, the coefficient of the x^{ 2} term must be 1.)
To achieve this we factor out 2. Then we can complete the square on the quadratic that remains inside the brackets. Here are the steps:
 We want the factor to be 2. The • quantities are yet to be determined.
 We use the distributive law to determine the • quantities inside the brackets.
 Now we are ready to complete the square on the quadratic inside the brackets.
Example 8: Express using exponential notation and factor out the lowest power of x.
Here are the steps:
 Rewrite the denominators using
negative exponents.
Notice 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.
 Optional. Use the negative exponent
property to rewrite the common factor as a fraction.
 Optional. Simplify by moving the
bracketted quantity into the numerator. Now we see that the result is
exactly the same as adding fractions
by finding a common denominator!
Example 9: In Example 5 we used the
GCF
(greatest common factor) of the denominators of as the denominator of the common factor. In this example we will instead use the
LCM
(lowest common multiple). Here are the steps:
 The LCM of
6 x^{ 2} and 9 x^{ 4} is 18 x^{ 4} and we put this in the factor’s denominator. The • quantities are yet to be determined.
 Determine the • quantities inside the brackets.
 Optional. Move the bracketted quantity into the numerator. Again we see that the result is
exactly the same as adding fractions by finding the LCD.
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