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.
Hint: It is always a good idea to expand or distribute out the factored form to check that you did the factoring correctly.


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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:
  1. Rewrite factors in denominators using negative exponents. This shows that the lowest power of x is −4.
  2. Factor out x to the lowest power. The • quantities are yet to be determined.
  3. Use the properties of exponents to determine the • quantities.
  4. Use the negative exponent property to rewrite the common factor as a fraction.
  5. 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:
  1. 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.
  2. Determine the • quantities inside the brackets.
  3. 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:
  1. 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.
  2. Determine the • quantities inside the brackets. The red and blue quantities match with the red and blue quantities in the original expression.
  3. 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:
  1. 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.
  2. 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.
  3. Use the negative exponent property to rewrite the common factor as a fraction.
  4. Simplify by moving the bracketted quantity into the numerator.

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