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Algebra Help: How to Multiply Rational Functions

When you need to multiply a rational function, the process is much like that of multiplying fractions, except with the use of the FOIL method. This post will explain the foil method and show how to use it to simplify rational functions.

The FOIL Method

The FOIL method is a way to keep track of a multiplication operation involving two binomials. It stands for FIRST, OUTSIDE, INSIDE, and LAST. When multiplying two binomials, (x+3) and (3x-4), you FIRST multiply the first elements within both sets of parentheses, in this case, x*3x=3x^2. Next you multiply the outside terms, x*-4=-4x. Then you multiply the INSIDE terms, 3*3x=9x, and then the LAST terms, 3*-4=-12. Combine them all together, and we get 3x^2+5x-12. Now let's apply the FOIL method to a rational function.


Step 1:

To solve this rational function, we first apply the FOIL method to the two numerators. (x+3)(3x-4)=3x^2-4x+9x-12 gives us 3x^2+5x-12.

Step 2:

Now do the same with the denominators. (2x-1)(5x+3)=10x^2+6x-5x-3=10x^2+x-3.

Step 3:

Put the two together and simplify as needed. Our result is: 

Example 2

Run FOIL on the numerators to get 2x^2+3x+2x+3=2x^2+5x+3. Then do the same on the denominators for x^2-x+3x-3=x^2+2x-3. Combine them and we get:    Let's look at one example that requires factoring.

Example 3

In this example , we first have to do some factoring. The first binomial can by expressed (x+2)(x-2). For the second numerator, we can factor it to (x-4)(x-3). And finally, the second denominator is x(x-2). Now our equation looks so: We can cancel out the two (x-2) and the two (x-3) and get Use FOIL on the numerators to get x^2-4x+2x-8=x^2-2x-8. Put the elements of the fraction into their designated places for Note that you can either express the result in standard form, as we have done here, or in factored form, which would be See the difference?

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