Reducing Rational Expressions – Polynomials EVERYWHERE!
A rational expression just refers to a fraction with a polynomial in the numerator, and a polynomial in the denominator.
Here are a few examples:
x^2 - 16
x + 4
x^2 - 2x - 8
x^2 - 9x + 20
4x + 4
x^4 - X^2
One thing that we need to keep in mind when working with rational expression is that divisibility by zero is not allowed. Just like when dealing with regular numbers. You cannot divide by zero. So, when dealing with a rational expression we will always assume that whatever x is, it will not give us division by zero.
Alright let’s take a look at how to reduce a rational expression. We are actually doing the same thing we would do when reducing a regular fraction.
So, let’s say we have 18 / 8. When we reduce this we can cancel our like terms. We can rewrite this as (9)(2) / (4)(2). We can cancel our two’s here giving us 9/4. 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 as (x-4)(x+4)/(x+4). Once we do this, we can see that our (x+4)’s cancel with each other, leaving us with (x-4).
Now, we need to be careful when canceling terms. The only reason we were able to cancel our (x+4)’s was because they are both being multiplied in the numerator and the denominator. This would not work if the top was (x-4) +(x+4) / (x+4).
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 gives us (x-4)(x+2)/(x-5)(x-4). We can go ahead and cancel our (x-4)’s from the numerator and denominator, which leaves us with (x+2)/(x-5). We now have it reduced to its lowest terms.
For our last example we have (4x+4)/ x^4 – x^2. To reduce it, we can rewrite our numerator by factoring out a 4. Giving us 4(x+1). In the denominator we can factor out an x^2, giving us x^2 -1, which we can factor even further to get (x-1)(x+1). So, now we have the rational expression 4(x+1)/x^2(x-1)(x+1). We can cancel out our (x+1)’s here, leaving us with 4/x^2(x-1). There is no further reduction we can do, so we now have it in its simplest form.
I hope that this video has been helpful to you.
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See you next time!