Reducing Algebraic Fractions Topic Index | Algebra Index | Regents Exam Prep Center

Monomial Fractions

If there is only one term (monomial) in the numerator and denominator, then common factors may be reduced directly.

Example:

 Reduce to lowest terms:

Solution:  Look at the "top" and the "bottom" of this fraction.  Find a common number that will divide evenly into both 36 and 24 (answer is 12).  Also, notice where the larger exponents of x and y are located.
It may help to expand the fraction to see what the exponents are doing.

After reducing by a common constant factor of 12 and
leaving "leftover" variables on the "top" and "bottom",
the

 Remember: There will always be a larger number (coefficient) "leftover" where there was a larger number "to begin with". Similarly, there will be variables "leftover" where there were "more" (larger exponents) to begin with. The "leftover" variables are determined by subtracting their exponents.

Polynomial Fractions

If there is more than one term in either the numerator or denominator (or both), you may need to factor.  Factoring will often allow for reducing, since factoring produces products and reducing can only occur when multiplication "connects" parts.  You can never reduce part of a sum or part of a difference.

 You cannot reduce part of a sum or part of a difference.

 This fraction can NOT be reduced.  The x in the numerator is "part of a sum" and can not be reduced by itself.

However,

 DOES reduce to This fraction can be reduced because of the multiplication in the numerator connecting the x to the x+5.  The first x in the numerator is NOT "part of a sum".

Example:

 Reduce

Solution:  At first glance, it may appear that since nothing is being multiplied, no reducing can occur.  But look more carefully!  If we factor first, we will be able to reduce this fraction.

NOW, after factoring, we can reduce the WHOLE binomial
(x + 5) since it is attached by multiplication.  We did not reduce "part" of the sum,
we reduced the entire sum.

Example:

 Reduce

 First, factor the top and bottom.

Reduce the binomial (x + 3) in the top and bottom.

Example:

 Reduce

Solution:  Factor the top and the bottom completely.

Example:

 Reduce

Solution:  At first glance, this fraction may look like it can not be reduced.
But look more carefully.  If -1 is factored out of the denominator, it will
be possible to reduce this fraction.