Hotmath
Math Homework. Do It Faster, Learn It Better.

Simplifying Rational Expressions

As you know, a rational number is one that can be expressed as a fraction , that is,

p q ,

where p and q are integers (and q ≠ 0 ).

Similarly, a rational expression (sometimes called an algebraic fraction ) is one that can be expressed as a quotient of polynomials , i.e. p q where p and q are polynomials (and q ≠ 0 ).

Example 1:

3 x 3   +   5 x 2 y   −   7 y 3

is a rational expression, since both the numerator and the denominator are polynomials . (" 3 " counts as a polynomial... it's just a very simple one, with only one term.)

5 x   +   3 6 x   +   x   −   x y

is not a rational expression. The denominator is not a polynomial.

A rational expression can be simplified if the numerator and denominator contain a common factor .

Example 2:

Simplify.

3 x   +   6 9 x 2   −   9 x   −   54

First, factor out a constant from both numerator and denominator. Write the 9 as 3 ⋅ 3 .

3 x   +   6 9 x 2   −   9 x   −   54 = 3 x   +   6 3 ⋅ 3 ( x 2   −   x   −   6 )

Next, factor the quadratic in the denominator. (Look for two numbers with a product of − 6 and a sum of − 1 .)

= 3 ( x   +   2 ) 3 ⋅ 3 ( x   +   2 ) ( x   −   3 )

Finally, cancel common factors.

= 1 3 ( x   −   3 )

IMPORTANT NOTE: EXCLUDED VALUES

When we factored out x + 2 in the above expression, we made an important change. The new expression

1 3 ( x   −   3 )

is defined for x = − 2 ; it equals − 1 5 . But the original expression we were trying to simplify,

3 x   +   6 9 x 2   −   9 x   −   54

is undefined for x = − 2 , because the denominator equals zero (and division by zero is a no-no).

So, our simplification is not really true for all points. When you simplify rational expressions, you should make note of these excluded values .