 So rational expressions are just like fractions, so we can add fractions, and we can also subtract them. So remember, you subtract fractions with the same denominator by subtracting the numerators. And if you have different denominators, you can always convert one fraction into a fraction with a different denominator by multiplying numerator and denominator by some factor. Meanwhile, the rules for subtracting rational expressions are radically different. If we want to subtract two rational expressions with the same denominator, we'll subtract the numerators. Well, that's not too bad actually. There is one important difference. Because the numerator of the rational expression could be a polynomial, we want to make sure we're subtracting the entire polynomial. So it's best to think about this subtracted term as being inside a set of parentheses. And again, if our denominators aren't the same, we could multiply numerator and denominator by some factor to change the denominator. For example, if we want to subtract 5 over 2x minus 3 over x squared, as with fractions, we have to get a common denominator. And the thing to remember is that the product of the denominators is a common denominator. So our rational expressions are 5 over 2x and 3 over x squared. So a common denominator is going to be 2x times x squared. So in our first fraction, the factor we're missing is x squared. So we'll multiply numerator and denominator by x squared to get our second fraction has a denominator of x squared. If we want that denominator to be 2x times x squared, then we need to multiply the denominator by 2x. And we also need to multiply the numerator by 2x to get us the fraction. And so my new equivalent fractions are 5x squared over 2x times x squared and 3 times 2x over 2x times x squared. So remember factored form is best, so let's leave our fractions in this form until we have to factor them or do something else. So now our rational expressions have the same denominator, so we can subtract the numerators. Now we might want to try and reduce this, but remember you can only remove common factors if you have a product. And in the numerator we don't have a product, we have a difference. So we have to expand this out, so now let's multiply 3 by 2x. And we'll sit and stare at this numerator 5x squared minus 6x and try to factor it. And now we have a product x times 5x minus 6, so now we can remove common factors. There's a factor of x in the numerator, there's a factor of x in the denominator, and we can remove that leaving as our final answer. Or let's take a more complicated subtraction. So again we can always find a common denominator by multiplying the two denominators together. So our denominators are x minus 6 and x plus 5, so a common denominator will be. And remember factored form is best, so let's leave our denominator in this product form. Now to make 8x over x minus 6 have a denominator of x minus 6 times x plus 5. We need to multiply numerator and denominator by x plus 5. And since factored form is best, we'll leave this in factored form. For our second fraction, we need to multiply numerator and denominator by x minus 6. And again we'll leave the numerator and denominator in factored form until we need to do something else. And now our denominators are the same, so we can subtract the numerators and keep the same denominator. And let's see if we can simplify this. We could try to cancel the x plus 5, which is in numerator and denominator, or the x minus 6, which is in the numerator or the denominator. But we would be wrong. And the important thing to remember is that you can only remove common factors if you have a product. And the numerator is not a product. It's a difference. And that means if we have any hope of simplifying this, we'll have to clear up this difference. So let's expand our terms out and simplify. And now we have something we might try to factor. But remember factoring is the hardest easy problem in mathematics, so let's see if we can make things a little bit easier. So remember a factor only matters if it's a common factor. Our denominator is still in factored form, and the factors are x plus 5 and x minus 6. And what this means is that we only care if x plus 5 or x minus 6 is a factor of the numerator. And so we check. Can we write our numerator as a product of x plus 5 times something? To begin with, since we want our product to give us an 8x squared, to get an 8x squared, we need the second factor to have an 8x term. And we also need this product of the constant 5 times a to be equal to 18. But this can't happen over the integers, so this won't give a factorization. What about the other factor? Can we write our numerator as x minus 6 times something? And again, we need to have an 8x term in that second factor. And this time we want minus 6 times a to be 18. So the only possibility is for a to be equal to negative 3. But we have to check. Can we write our numerator as x minus 6 times 8x minus 3? So we multiply out the right-hand side, but this is false. So this is not a factorization. And the important thing to recognize here is that even if 8x squared plus 37x plus 18 factored, we wouldn't be able to simplify our expression any further. And that means there's no point in even trying to factor this, because we can't get a simpler expression, so we'll leave it alone.