 So how can we add rational expressions? So it's useful to remember that rational expressions are to polynomials what fractions are to the whole numbers. So let's consider what happens when we add two fractions. In order to add two fractions, you need a common denominator. But if you don't have one, you can convert the fractions into equivalent fractions by multiplying numerator and denominator by the same thing. Well, that's arithmetic. Algebra is generalized arithmetic and so we can generalize this for rational expressions. If we have two rational expressions with the same denominator, we can add the numerators and keep the common denominator. And given any rational expression, we can multiply numerator and denominator by the same thing without changing what the expression actually is. So let's say we want to add 5 over x plus 2 over 3x. Our denominators are x and 3x, which means that 3x will be our common denominator. So we'll want to change both fractions into fractions with a denominator of 3x. So this fraction 2 over 3x will change so it has a denominator of 3x. To make 5 over x into a rational expression with denominator 3x, we need to multiply numerator and denominator by 3. And so if we do that, we get the equivalent fraction 15 over 3x. And now our denominators are the same, 3x, and so we can add the fractions by adding the numerators and keeping the common denominator. And we can simplify that numerator a little bit. 15 plus 2 gives us 17 over 3x. Or we could add x plus 3 over x plus 7 over x squared. Our denominators are x and x squared, also known as x times x. And so that means both of these can be changed into fractions with denominators of x squared. 7 over x squared becomes 7 over x squared. To make x plus 3 over x into a rational expression with denominator x, we need to multiply numerator and denominator by x. And for now we'll leave this in the form of x times x plus 3 because remember, factored form is best. Now both fractions have a denominator of x squared, so we can add the numerators. And even though factored form is best, the numerator is not in factored form because it's no longer a product. It's actually a sum. And what this suggests is we should expand the numerator and that gives us the numerator x squared plus 3x plus 7. How about something like x over x plus 3 plus 2 over x minus 4? Since our denominators are different, we need to find a common denominator. And a useful thing to remember is that the product of the denominators is a common denominator. And so a common denominator will be x plus 3 times x minus 4. So if I want x over x plus 3 to have this denominator, I'll need to multiply numerator and denominator by x minus 4. And if I want 2 over x minus 4 to have this denominator, it's missing that factor of x plus 3. So we'll multiply numerator and denominator by x plus 3. And remember factored form is best, so we'll leave our denominator in the form x plus 3 times x minus 4. And since the denominators are the same, we can add the numerators. And again, while factored form is best, our numerator is not in factored form because it's not a product. And so we'll expand that out and collect like terms, giving us our numerator. Our denominator is written as a product and factored form is best. We'll leave it in this form. So we might see if we can remove a common factor. But remember, a factor only matters if it's a common factor. And what this means is we only care of x minus 4 or x plus 3 is a factor. So we might see if x minus 4 is a factor of the numerator if we can write our numerator x squared minus 2x plus 6 as x minus 4 times something. But this would require minus 4b to be 6, but this has no integer solutions, so we can't get a factorization this way. We might also see if x plus 3 is a factor if we can write x squared minus 2x plus 6 as x plus 3 times something. And in this case, we need 3a to be 6, so the only possibility is a equal to 2. But we still don't know if we actually have that factorization, so let's check it out. Is this really x plus 3 times x plus 2? So if we expand the right-hand side, we see that this is not the correct factorization, so we can't factor this way either. Now, we could try to factor x squared minus 2x plus 6 as x plus a times x plus b, but we don't need to. Remember that a factor only matters if it's a common factor, because that's the only time we'll be able to remove it, and we've already determined that neither of our possibilities for a common factor is actually a factor. And what that means is we can't simplify this expression any further, and we'll leave it alone.