Rational Expressions

“Rational Expressions” are simply algebraic fractions.  Simplifying rational expressions means to make the fraction “look” more simple.  The key is that the problem does not have an equal sign ( = ).  Problems involving ( = ) prompt us to “solve”.

Simplifying rational expressions may involves reducing fractions, multiplying or dividing fractions, and adding or subtracting fractions.  All of these operations on fractions will involve polynomials and will use factoring techniques.

Reducing fractions (i.e. writing the fraction in “lowest terms”)
First, to reduce a fraction, we “divide” out common terms.  Many books use the word “cancel” to describe the action, but it is really division.
Think about this, to simplify (or reduce) the fraction  , we would first “factor” the 10 and the 16.  The result would look something like this,  . Since the numerator and denominator both

contain a common factor … which is a word associated with multiplication, then we could “divide”
the and get the result of    as the simplified or “reduced” fraction.

Now, apply this same procedure to fractions involving polynomials.  For example, let’s write the following fraction in “lowest terms.”

                   … first we must factor both the numerator and the denominator

               … now we observe that there is a common factor, namely  

        or        … divide out the common factor and we get the fraction in “lowest terms”

Notice that first I left the “terms” of the fraction are left within the (  ).  It is a common error to try and divide out the terms of the fraction.  If you leave the numerator and denominator within the (  ) there is less danger in making this mistake.  You may also drop the parenthesis after you have reduced the fraction.

Even though I removed the (  ), I cannot simplify any further.  Removing the (  ) turns the factors into “terms” and “terms” cannot be divided!

The LCD is used to make equivalent fractions, to add or subtract fractions with unlike denominators, and to simplify complex fractions.  The LCD is also useful in solving equations that contain fractions.

To find the LCD between two polynomial denominators, factor each denominator completely.  The LCD is the product of the factors of the polynomials to the greatest number of times that the factor is present in any one of the denominators.
For example, find the LCD between and  .  15 = (3)(5) and 18 = (2)(3)(3), so the

LCD = (2)(3)(3)(5) = 90.  This means that 90 is the least (smallest) number that can be used to make equivalent fractions of  and    so that they may be combined by addition or subtraction.

Notice that the (3) factor was used twice in the LCD since it was found the greatest number of times (twice) in 18.

Now, expand this understanding to polynomial denominators.  Find the LCD between

        and                … factor the denominators and we get

      and               …observe the factors of the denominators

The LCD = y(y – 5)(y + 3).  Notice we only use the (y – 5) once since it is only found once in each of the denominators.

Now suppose that we wanted to make equivalent fraction using the LCD.
      and 

It is best to leave the denominators factored so that we can observe the factor that is necessary to transform the original fraction into the new fraction with the LCD.

The first fraction is missing the (y + 3) factor, so we need to multiply the original numerator of 8 with the factor (y + 3).  Likewise, the second fraction is missing the (y) factor, so we need to multiply the original numerator of -2 with the factor (y).

       

      ... fractions now have a common denominator

Whenever you add or subtract fractions, you must have a common denominator.  If we now wanted to add the fractions above we would simply add like terms in the numerators and retain the common denominator.