Math

Simplifying Rational Expressions

As we mentioned before, a rational expression is a ratio which has polynomials in both numerator and denominator. When we simplify a numerical fraction, we have to divide both numerator and denominator by their common numerical factor. When we deal with rational expressions, we divide both numerator and denominator by their common numerical or variable factor.

Example: Simplify .

Step 1) Check for common numerical factors: the numerical factor in the numerator is 4 and in the denominator is 12. The least common factor of 4 and 12 is 4. Therefore, we divide both 4 and 12 by 4:

Step 2) Check for common variable factors: both numerator and denominator share. So, we divide both numerator and denominator by:

Example: Simplify .

Step 1) Check for common numerical factors: the lowest common factor between 5 and 15 is 5. Therefore, we divide both numerator and denominator by 5:

Step 2) Check for common variable factors: in order to simplify the fraction, we expand the numerator first:

we can simplify this even further using , hence

.

We use the above expression for and simplify the expression:

.

Example: Simplify .

Step 1) Factor numerator and denominator: we can factor in the numerator and in the denominator:

Step 2) Check for common numerical/variable factors: the factors and  differ in a negative sign. Therefore, we factor -1 from top or bottom and simplify: