How to Simplify Rational Expressions: Step-by-Step

by Tamara Runzel

Math is a complex subject for many students. You learn about a number of different processes, and some are difficult to understand. One area you will learn about in algebra is simplifying rational expressions. A rational expression is an expression that you can write as a fraction where the numerator and denominator are both polynomials and the denominator does not equal zero. Polynomials are terms you can add, subtract or multiply. A polynomial can have constants, variables and exponents. To simplify a rational expression, you cancel out factors that the numerator and denominator have in common.

Set up two groups of parentheses to factor the numerator and denominator of the rational expression. Before you can simplify an expression, you must factor the problem.

Place the appropriate factors in the first position in each set of parentheses. Use the example, (x^2 + 7x + 12)/(x^2 + 5x + 6). Your first step would show: (x )(x ) on the top and bottom because to have x^2, you would multiply x by x.

Find the factors that would go in the last position of your parentheses. The polynomial x^2 + 5x + 6 is said to be written in the form x^2 + bx + c. The product of your factors must equal "c" while their sum equals "b." If "c" is positive, your factors will have the same sign, depending on the sign for "b." If "c" is negative, your factors will have the opposite sign, with the greater number having the same sign as "b." In the example, (x^2 + 7x + 12)/(x^2 + 5x + 6), the factors for the numerator are 4 and 3 because multiplied they equal 12 and added they equal 7. The factors for the denominator are 3 and 2 because multiplied they equal 6 and added they equal 5.

Set up your expression to reflect the factoring. Instead of (x^2 + 7x + 12)/(x^2 + 5x + 6), your expression now shows (x+3)(x+4)/(x+3)(x+2).

Cross out any factors the numerator and denominator have in common. In the example, (x+3)(x+4)/(x+3)(x+2), you would cross out the factor (x+3) on the top and bottom. You are now left with (x+4)/(x+2).

Look for any other common factors you can cross out. The example, (x+4)/(x+2), is simplified. You cannot cancel the "x's" or the "4" and "2" because they are terms or part of the contents of a pair of parentheses.


  • Depending on the type of polynomial, you may use a different process to factor. You can use grouping or finding the greatest common factor, as well as the reverse FOIL method described above.

About the Author

Tamara Runzel has been writing military, parenting, family and relationship articles since 2008. Runzel started in television news, followed by education before deciding to be a stay at home mom. Her articles have appeared in military publications as well as numerous online publications. She holds a Bachelor of Arts in communication from University of the Pacific.

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