Simplifying Rational Expressions

We've already shown that a rational number is any number that can be expressed as the ratio between two integers. This relationship is frequently depicted as a fraction $\frac{p}{q}$ where both p and q are integers, and q does not equal 0. Irrational numbers are the opposite in that they cannot be expressed as a ratio between two integers or as a repeating decimal.

A rational expression is similar to a rational number in that it's any expression that can be depicted as a ratio of polynomials instead of integers. We still use the p/q format and q can't be equal to zero, but we're talking about the quotient of polynomials with the possibility of variables and exponents instead of individual numbers. It's a little bit more complex as a result.

For example, $\frac{3}{x^3+5x^{2}y-7y^3}$ is a rational expression because both the numerator and denominator are polynomials. Yes, 3 is a polynomial, albeit a simplistic one with only one term. Polynomials can have any number of terms.

However, $\frac{5x+3}{6x+\sqrt{x}-x^y}$ is not a rational expression because the denominator is not a polynomial. Remember that polynomials can only have positive integer exponents and cannot have any square roots. Now that you understand what a rational expression is, let's move on to how we can simplify them and make them easier to work with.

Simplifying rational expressions through factoring

A rational expression can be simplified if the numerator and denominator contain a common factor. Let's simplify a rational expression as an example:

$\frac{3x+6}{9x^2-9x-54}$

First, factor out a constant from both the numerator and denominator. This becomes easier if we write the 9 as $3\left(3\right)$ :

$\frac{3x+6}{3\left(3\right)\left(x^2-x-6\right)}$

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

$\frac{3\left(x+2\right)}{3\left(3\right)\left(x+2\right)\left(x-3\right)}$

Finally, we need to cancel any common factors. In this case, that's a 3 and $\left(x+2\right)$ . The final answer is:

$\frac{1}{3\left(x-3\right)}$

With so many steps involved, the best thing to do is to avoid rushing. Go one step at a time to avoid small mistakes and always remember to check your work. You'll probably be writing small numbers and symbols, so make sure your work is legible as well.

Your expression must remain rational when simplifying rational expressions

When we factored out $\left(x+2\right)$ above, we made an important mathematical change that you may not have noticed. The simplified version of this expression is defined for $x=-2$ , it equals $-\frac{1}{15}$ . However, if we substitute -2 for x in the original denominator, we get:

$9{\left(-2\right)}^2-9\left(-2\right)-54$

$36-\left(-18\right)-54=0$

That means we're dividing by zero, a big no-no that becomes even worse when we specifically defined a rational expression as $\frac{p}{q}$ when q does not equal 0 above. This means that any value of x that makes the original denominator equal to zero is not in the domain of the expression. You should make note of any excluded values whenever you simplify a rational expression.

The best way to identify any excluded values is to set the polynomial in the denominator equal to zero and attempt to solve for x. How you go about doing that depends on the numbers you're working with, as sometimes factoring by grouping makes the most sense while the quadratic formula is better in other instances. You might even be able to solve for x by completing the square.

Regardless of how you do it, remember that your simplified rational expression might have one or more values for x that would also lead to dividing by zero. If so, that would be an excluded value as well. You can note any excluded values using the (does not equal) symbol that we used to define a rational expression above. Of course, there may not be any excluded values to find either.

Simplifying rational expressions practice problems

a. Is $2x+\frac{4}{3}y{z}^2$ a rational expression? Why or why not?

Yes, both the numerator and denominator are polynomials

b. Is $\frac{7x+1}{6x<msup>\; <mn>-3</mn>\; </msup>}$ a rational expression? Why or why not?

No, the denominator has a negative exponent.

c. Simplify: $\frac{x^2+x-2}{3x^2+9x+6}$

$\frac{x-1}{3\left(x+1\right)}$

d. Are there any excluded values from your answer above?

Yes, both -2 and -1 yield a denominator of zero in the original equation

Topics related to the Simplifying Rational Expressions

Dividing Rational Expressions

Graphing Scale and Origin

Adding and Subtracting Rational Expressions with Unlike Denominators

Flashcards covering the Simplifying Rational Expressions

Algebra II Flashcards

College Algebra Flashcards

Practice tests covering the Simplifying Rational Expressions

Algebra II Diagnostic Tests

College Algebra Diagnostic Tests

Varsity Tutors can help with simplifying rational expressions

Simplifying rational expressions is an essential building block toward performing operations on rational expressions, meaning that your student will have a hard time moving forward if they're not sure what their math teacher is doing in class right now. Fortunately, this is not something your learner has to try and figure out on their own. An experienced math tutor could look for fresh ways to explain the material while providing a distraction-free learning environment to help your student learn more effectively. Reach out to the friendly Educational Directors at Varsity Tutors for more information.