Before you start simplifying or otherwise manipulating rational expressions, take a moment to review what the rational expression itself is: A fraction with a polynomial in both the numerator and the denominator. Or, to put it another way, a ratio of one polynomial to another. Once you've identified a rational expression, the process of simplifying it boils down to three steps.
The Steps in Simplifying Rational Expressions
The process for simplifying rational functions follows a fairly simple roadmap. The first thing you must do is combine like terms, if you haven't already, to help you see the polynomials clearly.
Next, factor each polynomial. Sometimes all you have to do is write out every term. For example, it's clear that 4x (which is in fact a polynomial, even though it has only one term) has two factors: 4 and x. But with more complicated polynomials, your best tool is often recognizing patterns for specific types of polynomials you've already learned about. For example, if you've been paying close attention to your formulas, you might remember that a polynomial of the form a2 - b2 factors out to (a + b)(a - b).
Once your polynomials are fully factored, the last step is canceling any common factors that appear in both the numerator and the denominator. The result is your simplified polynomial.
What if the polynomials in your rational expression aren't of a form that you know how to easily factor? There are other techniques you can use to factor them, such as completing the square or using the quadratic formula.
A Warning About the Denominator
You might not be surprised to hear that there's a little catch here. Usually the domain (or set of possible x values) for your rational expression are assumed to be the set of all real numbers. But if anything happens to make the denominator of your fraction zero, the result is an undefined fraction.
What would make your denominator zero? Usually a little examination is all it takes to find out. For example, if the denominator of your fraction has been reduced to the factors (x + 2)(x - 2), then then the value x = -2 would make the first factor equal to zero, and x = 2 would make the second factor equal to zero.
So both of those values, -2 and 2, must be excluded from the domain of your rational expression. You'll usually notate this with the "not equal" sign or ≠. For example, if you need to exclude -2 and 2 from the domain, you'd write x ≠ -2, 2.
Simplifying Rational Expressions: Examples
Now that you understand the process of simplifying rational expressions, it's time to look at a couple of examples.
Example 1: Simplify the rational expression (x2 - 4) / (x2+ 4x + 4)
There are no like terms to combine here, so you can skip that first step. Next, with your keen eyes and a little practice, you can spot that the numerator and denominator are both easily factored:
(x + 2)(x - 2) / (x + 2)(x + 2)
Perhaps you'll also spot that (x + 2) is a factor in both the numerator and the denominator. Once you cancel the shared factor out, you're left with:
(x - 2) / (x + 2)
You've simplified your rational expression as far as you can, but there's one more thing to do: Identify any "zeroes" or roots that would result in an undefined fraction, so you can exclude those from the domain. In this case, it's easy to see by examination that when x = -2, the factor on the bottom will equal zero. So your simplified rational expression is actually:
(x - 2) / (x + 2), x ≠ -2
Example 2: Simplify the rational expression x / (x2 - 4x)
There are no like terms to combine, so you can go straight to factoring by examination. It's not too hard to spot that you can factor an x out of the bottom term, which gives you:
x / x(x - 4)
You can cancel the x factor from both numerator and denominator, which leaves you with:
1 / (x - 4)
Now your rational expression is simplified, but you also need to note any x values that would result in an undefined fraction. In this case, x = 4 would return a value of zero in the denominator. So your answer is:
1 / (x - 4), x ≠ 4
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