11.1 - Simplification of algebraic fractions

Some definitions



Division by zero

This operation is not allowed in mathematics. Click here to see why. This means that in the algebraic fraction
,
x cannot equal 1 or −3 because those values of x would cause the fraction to have a denominator of zero.



Reducing an algebraic fraction to lowest terms

Look at the expressions to the right. Note the following:
If we go in the reverse direction then we say that we are reducing a fraction to its simplest equivalent fraction or to lowest terms. To accomplish this we find any factor that is contained in both the numerator and denominator and cancel it out or strike it out, like this:



Example: Reduce the common fractions 10/6 and 10/5 to lowest terms.
Factor the numerator and denominator. Cancel the common factor of 2.
Factor the numerator and denominator. Cancel the common factor of 5. The result of the division is an integer. We say that the denominator divides evenly into the numerator.



If the numerator and denominator of an algebraic fraction are both monomials, then take all of the following steps to reduce the fraction to lowest terms:


Example:  Reduce the algebraic fraction to lowest terms.
Solution:
The − sign is put either in front of the result or in front of the numerator; never in front of the denominator.

Reduce the coefficient 6/9 to lowest terms.




Example:  Reduce the algebraic fraction to lowest terms.
Solution:
The two − signs are replaced by a + sign which we don’t have to display. The coefficient reduces to ¼. The numerator contains other factors so the 1 in the numerator can be omitted.

Combine the exponentials with base x using the properties of exponents.



Example:  Reduce the algebraic fraction to lowest terms.
Solution:
The − sign is put in front. The coefficient reduces to 1/3. The identical factors of x 3 in the numerator and denominator cancel. The numerator contains no other factors so this time the 1 must be displayed.




Example:  Reduce the algebraic fraction to lowest terms.
Solution:
After carrying out all the simplifications, the denominator equals 1, so we don’t have to display it. Thus the result is an ordinary expression, not an algebraic fraction.



If the numerator and denominator of an algebraic fraction are both multinomials, then in addition to the steps listed above, try the following steps to reduce the fraction to lowest terms: In the following examples we will assume that you already know how to do the factoring so we will just show how to use the factors to reduce the algebraic fractions to lowest terms.




Example:  Reduce the algebraic fraction to lowest terms.
Solution:
Factor the numerator and denominator.


Cancel the common factor of x.







Example:  Reduce the algebraic fraction to lowest terms.
Solution:
Factor the numerator.


Cancel the common factor of x − 2.






Example:  Reduce the algebraic fraction to lowest terms.

Solution: This is the same algebraic fraction as in the previous example except that the denominator differs by a − sign.

Factor the numerator and factor a − sign out of the denominator.


Cancel the common factor of x − 2.


Bring the − sign to the numerator and distribute it.




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