Reducing Rational Expressions – Polynomials

Hey guys! Welcome to this video on simplifying rational expressions.

What is a Rational Expression?

A rational expression just refers to a fraction with a polynomial in the numerator, and a polynomial in the denominator.

Rational Expression Examples

Here are a few examples:

1. $\frac{x^2-16}{x+4}$
2. $\frac{x^2-2x-8}{x^2-9x+20}$
3. $\frac{4x+4}{x^4-x^2}$

One thing that we need to keep in mind when working with rational expression is that divisibility by 0 is not allowed. Just like when dealing with regular numbers, you cannot divide by 0. So, when dealing with a rational expression, we always assume that whatever $x$ is, it will not give us division by 0.

How to Reduce Rational Expressions

Alright let’s take a look at how to reduce a rational expression. We’re actually doing the same thing we would do when reducing a regular fraction.

Example #1

So, let’s say we have $\frac{18}{8}$. When we reduce this, we can cancel our like terms. So we can rewrite this as:

 
We can cancel our 2s here giving us:

 
So now we have a fraction reduced down to its simplest form. There is not another number that both our numerator and denominator are divisible by.

It works the same way with a rational expression.

Let’s try reducing our first example.

 
We can rewrite our numerator, once we factor this out, as:

 
And once we do this, we can see that our $(x+4)$s will cancel out. So we cancel that out, leaving us with $x-4$

Now, we need to be careful when canceling terms. The only reason we were able to cancel out our $(x+4)$s here was because they are both being multiplied in the numerator and the denominator. This would not work if our top was: $\frac{(x+4)+(x-4)}{(x+4)}$.

Example #2

Let’s now move on to our second example, which is a bit trickier.

 
We can do the same thing that we did in our first example by rewriting our numerator and denominator. So that would give us:

 
So, we can go ahead here and cancel our $(x-4)$s, which would leave us with:

Example #3

For our last example we have:

 
To reduce it, we can rewrite our numerator by factoring out a 4, which would give us $4(x+1)$. In the denominator we can factor out an $x^2$, which would give us $x^2(x^2-1)$.

 
But, notice, we can factor this out even further so we can get something to cancel out with our numerator here.

 
At this point, we can cancel out our $(x+1)$s here, leaving us with:

 
There is no further reduction we can do, so we now have it in our simplest form.

I hope that this video has been helpful for you!

See you guys next time!

Rational Expression Practice Questions