11.1 - Simplification of algebraic fractions

Some definitions

A common fraction is a number that is written in the form or a/b, where a, the numerator, and b, the denominator, are both integers. A common fraction is used to describe a part or fraction of a whole object. The notation means that we break an object into b equal parts and we have a of those parts. The portion or fraction of the object that we have is a/b.
Division is defined in terms of multiplication. Dividing a number a by a number b produces a number c such that c multiplied by b gives back a. We use the same fraction notation, a/b, to denote the division of a by b because when a and b were both integers, then the division a/b results in the common fraction a/b.
An algebraic fraction is one whose numerator or denominator are algebraic expressions. Two examples of algebraic fractions are
and .
A rational algebraic fraction is an algebraic fraction whose numerator and denominator are both polynomials. The first example above is a rational algebraic fraction; the second one is not.
A proper common fraction is a common fraction whose numerator is smaller than its denominator, and an improper common fraction is one whose numerator is greater than or equal to its denominator. A mixed fraction is the sum of an integer and a proper fraction. Long division can be used to convert an improper fraction to a mixed fraction.
A proper algebraic fraction is a rational algebraic fraction whose numerator is of lower degree than its denominator, and an improper algebraic fraction is one whose numerator is of greater or equal degree than its denominator. A mixed expression is the sum of a polynomial and a proper algebraic fraction. Long division can be used to convert an improper algebraic fraction to a mixed expression.

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:

We start with the fraction a/b.
Multiply it by 1. This will not change its value.
Write “1” as the fraction d/d.
Multiply the two fractions. The numerator of the new fraction is ad and the denominator is bd.
The final fraction is equivalent to the first fraction.

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:

Obtain the sign by using the rules for signs.
Reduce the coefficient to lowest terms.
Cancel identical factors that appear in both the numerator and denominator.
Combine exponentials having the same base by using the division property of exponentials.

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 x³ 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:

Factor the numerator or denominator or both. Sometimes this will cause new canceling factors to appear.
Factor a − sign out of the numerator or denominator. Sometimes this will cause a new canceling factor to appear.

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.

Algebra Coach Exercises