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Simplifying Rational Expressions

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Today we're going to talk about simplifying rational expressions. So we'll start by reviewing how to simplify numerical fractions. And then we'll do some examples simplifying rational expressions.

So let's start by reviewing how to simplify or reduce a numerical fraction. I can see what this fraction might reduce to by breaking it down into its factors.

So I know that 18 is equal to 2 times 9. And I can break down 9 further by writing that as 3 times 3. 4 I can break down by writing as 2 times 2.

So now that I have both the numerator and the denominator broken down into factors. I can cancel out or cross out any common factor in the numerator and the denominator.

So I see that I have a 2 in the numerator and the denominator. So those can cancel out. Then I can just simplify what I have left. 3 times 3 will give me 9. And then I have 2 in my denominator. So 18 over 4 reduces or simplifies to 9 over 2.

So let's do an example simplifying a rational expression. I have x plus 7, times x plus 4, over x minus 1, times x plus 7. I notice that both the numerator and the denominator are already in factored form. So then I can look for common factors between the numerator and the denominator that can be canceled out.

I see that I have x plus 7 as a factor in the numerator. And x plus 7 as a factor in the denominator. So I know that I can cancel those out. And I'm left with x plus 4 in my numerator. And x minus 1 in my denominator.

Now I don't have any more common factors that I can cancel out. We can't cancel out part of a factor if it has more than one term in the factors. So for example, even though I have an x term here and an x term here, x plus 4 is the entire factor. And x minus 1 is the entire factor.

So since the entire factors are not both the same, I can't cancel out part or the entire factor. So this is as simplified as this expression can be.

So here's another example simplifying a rational expression. I have negative 15x to the third plus 10x squared, over 10x squared minus 5x. So I need to write both the numerator and the denominator in factored form in order to see if any of the factors will cancel out.

So I see that in my numerator and in my denominator, they all have a common factor of 5x. So in the numerator, I'm going to factor out a 5x. And then my other factor will start with a negative 3x squared.

And then I'll have 2x on the inside. And that's because 5x times negative 3x squared will give me negative 15x to the third. And 5x times 2x will give me positive 10x squared.

I'll do the same thing in the denominator. Factor out a 5x. I already know that I'll need a 2x here so that when I multiply I'll get 10x squared. And here I'll just need a minus 1. Negative 1 times 5x will give me negative 5x.

So now that I have factored out 5x from both of them, since it's in the numerator and in the denominator, I can go ahead and cancel them out. And so now my expression becomes negative 3x squared plus 2x over 2x minus 1.

I don't have any more common factors in my numerator or between my numerator and my denominator. So this is as simplified as this rational expression can be.

So for my last example, I've got the rational expression, x squared plus 4x minus 21, over x squared, plus 8x, plus 7. To simplify, I know I want to write it in factored form so that I can see if any of the factors will cancel between the numerator and the denominator.

I know that x squared plus 4x minus 21 in factored form is going to be x plus 7, times x minus 3. And that's because positive 7 times negative 3 will give me negative 21. Positive 7 plus negative 3 will give me a positive 4.

Similarly, in the denominator, I can factor it as x plus 7 and x plus 1. And, again, that's because 7 times one will give me positive 7. 7 plus 1 will give me 8.

So now that I've written my numerator and denominator in factored form, I see that I do have two common factors, x plus 7 in the numerator and the denominator will cancel out. And I'm left with x minus 3 over x plus 1. And this is as simplified as my original expression can be.

So let's go over our key points from today. A rational expression is a fraction in which the numerator and denominator are polynomials. To simplify rational expressions, factor the expressions in the numerator and denominator. And cancel out any common factors.

This is the same process as simplifying numerical fractions. And in a fraction, you cannot cancel out part of a factor. Only entire factors can be canceled.

So I hope that these key points and examples help to understand a little bit more about simplifying rational expressions. Keep using your notes and keep on practicing. And soon you'll be a pro. Thanks for watching.