Hi and welcome. My name is Anthony Varela, and today we're going to simplify rational expressions. Well, rational expressions are also called algebraic fractions, so they are fractions that have variables in them. And to simplify rational expressions, we're first going to relate this to the process of simplifying numeric fractions, and then we'll follow that same process to simplify algebraic fractions. So let's go through an example of simplifying a numeric fraction.
So here I have 24 over 18. Now to simplify this fraction, I am going to identify all the factors in both the numerator and the denominator, and then I'll cancel out common factors, so what appears in both the top and the bottom of my fraction. So looking at our numerator, 24, and I'm going to break this down into all of its prime factors, so factoring it as much as I can. So 24 divided by 2 is 12, and if I divide by 2 again, that's 6. Dividing by 2 again is 3, and then I'll divide by 3 to get to 1. So these are my prime factors of 24. 2 times 2 times 2 times 3.
Let's do the same thing with 18. So 18 divided by 2 is 9. 9 is 3 times 3. So the prime factorization of 18 is 2 times 3 times 3. So now let's look at common factors. So what appears in both the numerator and the denominator? And so I've highlighted in red all of our common factors, so we're going to ignore them. They cancel out. So 24 over 18 simplifies to 2 times 2 divided by 3 or 4/3. So we're going to follow the same idea then to simplify a rational expression or an algebraic fraction.
So here we have x plus 2 times x minus 4 divided by x minus 4 times x plus 1. So these are already written out in factored form. So these are our factors. Everything grouped in parentheses that gets multiplied, those are our factors. So do we see any common factors? Well, in both the numerator and the denominator I see x minus 4, so that could be canceled. And what I'm left with then is x plus 2 over x plus 1. Now be careful.
This is a common mistake that many people make when they're first learning about simplifying rational expressions. We do not simplify this further to 2 over 1 because what we have is one single factor in the numerator, one single factor in the denominator, and we cannot cancel terms. So x and 2 are terms in the numerator, x and 1 are terms in the denominator. We cannot cancel the terms x and x and be left with 2 over 1. That is not equivalent, so be careful about making that mistake.
All right, now let's practice simplifying some other rational expressions. Here we have 3x squared plus 6x over 6x squared minus 3x. Now this is different from our example before is that this is not written out in factors, so we're going to see what we can factor in our numerator and denominator and see if we have anything in common. So looking at our numerator, well, I see that our two terms, 3x squared and 6x, they certainly share a factor of 3. 3 is a factor of 3 and 6, and they share a factor x. x is a factor of x squared and, of course, of x. So I can factor out 3x.
So in my numerator factoring out a 3x, I have x plus 2 in parentheses. All that's being multiplied by 3x, that's equivalent expression here. So now let's do the same thing to the denominator. Once again, they share a numeric factor of 3 and an algebraic factor of x, so I can factor 3x again. That's convenient. So here we have 3x times 2x minus 1. Now I have a common factor in both the numerator and the denominator that can be canceled out. So I have x plus 2 over 2x minus 1, and I can't cancel out the terms, so this is our expression simplified.
Let's go through one more example, x squared minus x minus 6 over x squared plus 3x plus 2. Well, I recognize both my numerator and my denominator as quadratic expressions that can be factored. So let's go ahead and factor the numerator, factor the denominator, and see if we have common factors. So with x squared minus x minus 6, I can factor this into something where one factor is x plus some value times x plus another value. And I'm looking for two integers where when we multiply them together, they equal negative 6, but we add them together, we get negative 1.
And those two values are going to be negative 3 and positive 2. We can check this with FOIL, and we will arrive at x squared minus x minus 6. Our denominator, once again, we're looking for two integers, but when we multiply them together, they're going to equal positive 2, and when we add them together, they're going to equal positive 3, the coefficient in front of the x term. And those two values would be positive 2 and positive 1. Once again, we could use FOIL to confirm that these two statements are equivalent. So I factored my numerator and I factored my denominator, so an equivalent expression here with them would be x minus 3 times x plus 2 over x plus 2 times x plus 1.
Now do we see any common factors? Well, I see an x plus 2 in both the numerator and denominator, so this simplifies then to x minus 3 over x plus 1. So let's review our lesson on simplifying rational expressions. The big idea is is that we want to identify all factors in our numerator and our denominator, and we want to cancel the common factors. So what appears on the top and the bottom of our fraction, but remember, we cannot cancel terms, so we're only canceling factors. So our strategy is we would factor out a common factor, so factoring our numerator completely, factoring our denominator completely, are the ways that we're going to identify common factors that we see in both the numerator and the denominator. So thanks for watching this tutorial on simplifying rational expressions. Hope to see you next time.