# Rational expressions

A rational expression is made of two polynomial expressions: the numerator (or top) and the denominator (or bottom), where the denominator is not the constant polynomial 0. Think of a rational expression as a fraction: the numerator divided by the denominator. Every polynomial may be interpreted as a rational expression whose denominator is the constant polynomial 1.

To evaluate a rational expression at a particular value of each variable, evaluate the numerator and denominator and divide the results. The rational expression is undefined whenever the bottom evaluates to 0.

## Arithmetic and simplifying

A rational expression is simplified if
• the top and bottom have only integer coefficients,
• the leading coefficient on the bottom is positive,
• the coefficients have no common integer factor greater than 1, and
• the top and bottom have no common polynomial factor that is not constant.
We can simplify rational expressions by factoring the top and bottom and cancelling any common factors; we may then leave the expression in factored form. (If the leading coefficient in the numerator is negative and we leave the expression in factored form, then we place the minus sign in front of the entire expression.)

We add, subtract, multiply, and divide rational expressions using the same techniques as for rational numbers. But systematic factoring is now more important, since less can be done by trial and error. When multiplying or dividing, you should factor all of the numerators and denominators, and there is no need to get common denominators; when adding or subtracting, you only need to factor the denominators at first, and you need to make all of these denominators the same. (You should still factor the numerator of the final answer when adding or subtracting, but that's only to simplify the result.)

A fraction whose top or bottom (or both) contains fractions is called complex. The straightforward way to simplify a complex fraction is to simplify the top and bottom separately and then divide them. Another way is to find a common denominator, not of just the fractions in the top or of just the fractions in the bottom, but of all of the fractions that appear in either; then you can multiply both top and bottom by this common denominator to turn the complex fraction into a simple one. (You'll still need to simplify this fraction, however.)

## Equations

To solve an equation between rational expressions, you can cross-multiply: multiply the top of one side by the bottom of the other side, then set the two resulting polynomials equal. In other words, to solve A/B = C/D, you solve AD = BC, or equivalently solve AD − BC = 0. If you are unable to factor this polynomial, then try factoring the original tops and bottoms; anything that shows up in both A or D and in B or C can then be factored out.

There is another method of solving rational equations. Here, you just factor the denominators, then multiply both sides of the equation by a common denominator. This is especially helpful when the equation has addition or subtraction of rational expressions; instead of simplifying each side (which also requires finding common denominators) you can multiply everything by a common denominator and make all of the fractions go away. (You still have to solve the resulting polynomial equation.)

Either way, you must check for extraneous solutions! Since every method of solving rational equations involves multiplying both sides by something that might be zero, you need to check that none of these expressions that you multiplied by actually evaluates to zero for any of your solutions. Equivalently, you can check that the original expressions in the equation are both defined for all of your solutions, with no division by zero. If any of your solutions fails this check, then you must throw it out. (Sometimes you'll throw out all of them, sometimes none of them; you never know until you check.)

Go back to the course homepage.

This web page was written between 2010 and 2016 by Toby Bartels, last edited on 2016 August 2. Toby reserves no legal rights to it.

The permanent URI of this web page is `http://tobybartels.name/MATH-1100/2016SU/rational/`.