How to Simplify Rational Expressions

Many of my students have struggled with how to simplify rational expressions, so I’ve decided to provide you with a few step-by-step examples of the main types you will see in class (and on tests 😉.) So, let's dive in!

1) Simplify:

So, we’ve got two fractions whose denominators are different. This is a problem, because we can only add or subtract fractions when their denominators are equivalent. Let’s rewrite the expression like this:

Now we know the denominator of the fraction on the right is missing an “x”, so we’re going to multiply that fraction by x (on the top and bottom):

Now, both fractions have the same denominator and we can just add the numerators:

Not so bad, right? Let’s try another..

Simplify:

If you notice, the first fraction has a denominator of (x+1) and the second has a denominator of x. This DOES NOT mean that the second denominator is only missing a 1. That’s super common to think, but you must look at the (x+1) as a whole factor and we can't just break that up in this case.

So, the left fraction is missing an x in the denominator, and the right fraction is missing an (x+1) and I’ll show you how we can fix this:

We’ve multiplied both fractions by the factors that were missing in the denominator. We’re not going to use the distributive property (although it's tempting, I know) so let’s rewrite our expression and see if it looks a little better:

As ugly as this may look, look at the denominators. They’re the same! (phew!!) Now we can add the numerators and combine both fractions into one fraction, just like the last example:

If you’ve noticed, our final answers look ridiculous (what does the above fraction even mean?!) That’s often the case with rational expressions so don’t be alarmed if you end up with a crazy looking fraction like above. With that out of the way, let’s try another!

Simplify:

If you don’t remember how to factor quadratic equations, you will want to review that before as rational expressions/equations do include them somewhat regularly.

I can see the denominator of the second fraction can be factored so let’s do that first:

Now, we see our left fractions’ denominator is missing an (x+4) (x+3) and our right fraction is missing a 2x. Let’s remedy that situation:

Once again, the denominators are the same (thank goodness!) and we can add the numerators:

Again, we have a crazy-looking answer, but that’s how we simplify rational expressions in a nutshell. Hopefully this was helpful, and I wish you the best of luck with your studies!