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.
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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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:
  1. One denominator is x2 and the other denominator is x4 and their common factor is x2. We put this in the factor’s denominator. The • quantities are yet to be determined.
  2. Use the properties of exponents to determine the • quantities.



Example 5:   Factor  

Here are the steps:
  1. 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.
  2. 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:
  1. 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.
  2. 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 completing-the-square 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:
  1. We want the factor to be 2. The • quantities are yet to be determined.
  2. We use the distributive law to determine the • quantities inside the brackets.
  3. 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:
  1. Rewrite the denominators using negative exponents. Notice 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. Optional. Use the negative exponent property to rewrite the common factor as a fraction.
  5. 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:
  1. 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.
  2. Determine the • quantities inside the brackets.
  3. 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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