Simplifying Rational Expressions

 

Learning Objective(s)

·         Simplify fractions with polynomials in the numerator and denominator by factoring and simplifying them.

 

 

Rational expressions are fractions that have a polynomial in the numerator or the denominator or both. Although rational expressions can seem complicated because they contain variables, they can be simplified in the same way that numerical fractions are simplified.

 

 

The first step in simplifying a rational expression is to determine the domain, the set of all possible values of the variables. The denominator in a fraction cannot be zero because division by zero is undefined. So we need to figure out what values of the variable(s) in the expression would make the denominator equal zero. These values cannot be included in the domain, so they're called excluded values. We discard them right at the start, before we go any further.

 

For rational expressions, the domain will exclude values for which the value of the denominator is 0. Two examples to illustrate finding the domain of an expression are shown below.

 

 

 

That wasn't hard. Let's try one that's a little more challenging:

 

 

 

Find the domain of the rational expression . A) all x B) all x not equal to 2 or 8 C) all x not equal to 0 D) all x not equal to -2, 2   Show/Hide Answer A. Correct. There are no values of x for which the denominator is equal to 0.  Therefore, there are no values that are excluded from the domain.   B. Incorrect. 2 and 8 do not make the denominator 0, therefore they are not excluded values. Set the denominator equal to 0 and solve for x.  This yields the excluded values, but in this case, there are none. The correct answer is all x.   C. Incorrect. If you substitute 0 for x in the expression, the denominator equals 8.  Only values that result in a denominator of 0 are excluded. Values that result in a numerator of 0 are not excluded. The correct answer is all x.   D. Incorrect. When the denominator is evaluated for x = -2 and x = 2, the denominator is not equal to 0.  So, these values are not excluded. The correct answer is all x.

 

 

Once we've figured out the excluded values, the next step is to simplify. To simplify a rational expression, we follow the same approach we'd use to simplify numerical fractions: find common factors in the numerator and denominator. Let’s start by looking at this method for a numerical fraction:

 

 

 

Now, we could have done that problem in our heads. But it was worth writing it all down, because that's exactly how we simplify a rational expression.

 

So let's simplify a rational expression, using the same technique we applied to that fraction just now. Only this time the numerator and denominator are both monomials with variables:

 

 

 

See—the same steps worked again. In the examples that follow, the numerator and the denominator are polynomials with more than one term, but the same principles of simplifying will once again apply. Factor the numerator and denominator to simplify the rational expression.

 

 

 

 

 

No matter how many terms or variables there are in a rational expression, we can simplify it by following the same steps:

 

The Steps for Simplifying a Rational Expression:   Determine the domain.  The excluded values are those values that result in a denominator of 0.   Find common factors for the numerator and denominator and simplify.

 

 

Simplify the rational expression below and state the domain.     A)  ; domain is all x   B)  ; domain is all x not –10 or   C)  ; domain is all x   D) ; domain is all x not –10 or   Show/Hide Answer A) Incorrect. You must first factor the polynomials in the numerator and the denominator and then express like factors in the numerator and denominator as 1 to simplify. You can only do this with factors, not terms in a polynomial. The domain is all x not –10 or  because these two values result in a denominator of 0. The correct answer is  ; domain is all x not –10 or .   B) Correct. The rational expression can be simplified by factoring the numerator as (2x + 3)(x + 5) and the denominator as (2x + 3)(x + 10) and then since  = 1, you can simplify the expressions to . The values x= –10 and x = give a denominator equal to 0, so the domain is all x not –10, .   C) Incorrect. The rational expression has been simplified correctly, but the domain is not all x. The values x= –10 and x = give a denominator equal to 0. The correct answer is  ; domain is all x not –10, .   D. Incorrect.  cannot be simplified to . The correct answer is  ; domain is all x not –10, .

 

 

Rational expressions, or fractions containing polynomials, can be simplified much like fractions can be simplified. To simplify a rational expression, first determine common factors of the numerator and denominator, and then remove them by rewriting them as expressions equal to 1.

 

An additional consideration for rational expressions is to determine what values are excluded from the domain. Since division by 0 is undefined, any values of the variables that result in a denominator of 0 must be excluded. Excluded values must be identified in the original equation, not its factored form.