 So let's take a look at something called a rational expression, and a rational expression is the quotient of two polynomials. So for example, 3x plus 5 over 2x minus 7, it's a quotient of 3x plus 5 divided by 2x minus 5. Both of these are polynomials, so that's a rational expression. Likewise, x squared minus 7x plus 8 over 3x minus 5, this is a quotient of two things, both of which are polynomials, so this is a rational expression. x plus 7 over 3 over x minus 7, well, it is a quotient. It is a quotient of two things, but one of the things is not a polynomial, so this is not a rational expression. Now, I'll add a small qualifier here. It is possible. It is conceivable that we can do some sort of algebraic simplification on this and produce a rational expression, but as it's written, this is not a rational expression, and it's a philosophical point, but as it's written, this is not a rational expression, even though in potentia, this could be a rational expression. For us, we are, for this point in time, concerned with what is and not what could be. For what could be, that's when you take calculus. Well, again, algebra is a generalized arithmetic, so it's useful to think about rational expressions as being equivalent to fractions, and if you have a fraction, the one absolute rule that you have from arithmetic is that the denominator can never be equal to zero, and this is not ordinarily a problem. If I have a fraction like 3 over 5, I could say, well, we absolutely cannot allow 5 to be zero. 5 is never allowed to be zero, and well, that's not really a problem. But the problem arises when we look at rational expressions, because the denominator of a rational expression is a polynomial. It's possible that there might be some values of x that make the polynomial equal to zero, and these are the forbidden values, and we have to be careful that we identify these early. And so as a matter of habit, as a matter of something you automatically do when you are confronted with a rational expression, the first thing you really want to do is to identify the forbidden values and exclude them from consideration. So for example, let's take the rational expression, x squared minus 2x minus 5 over x squared plus 4x minus 12, and see what values of x we want to exclude. So we have to exclude any value of x that makes the denominator equal to zero. So a very common strategy is we see what makes this happen, and we forbid that from occurring. So I want to find where does denominator equal zero, and this is a quadratic equation, so we waste time trying to factor. We use the quadratic formula because we have a nice, fast, quick, efficient way of solving any quadratic. And I'll substitute my values of a equals 1, b equals 4, c equals negative 12. I substitute those into the quadratic formula, and after all the arithmetic does settles, I get my two solutions, x equals 2 or negative 6. Now, since x equals 2 or negative 6, solve this equation. x equals 2 or negative 6 will make the denominator zero, and I have to exclude them. I have to forbid that possibility. x cannot be 2 or negative 6, and I'll write that exclusion this way. Well, what about simplifying things? So a rational expression is like a fraction, and one of the rules I have for fractions, one of the important rules I have for fractions, is that if I have a common factor in numerator and denominator, then I can drop that common factor, and so if I can factor numerator and factor denominator, and if it happens that there's a common factor in both, I don't need to carry it over. Well, both are quadratic, so we should always use the quadratic formula to find the roots and then factor. So I need to find the forbidden values for the denominator anyway. I'm going to want to find the values that make this zero. I'm going to need to find the roots anyway, so it's not a great hardship. So here's my quadratic equation, 3x squared minus 11 plus 6 equals zero. Denominator equal to zero. I have my quadratic with a equals 3, b equals negative 11, c equals 6, and I'll substitute those values into the quadratic formula. I'll let the dust clear. I'll do some arithmetic. I'll do some arithmetic. I'll do some arithmetic. I'll do some arithmetic, and I get my two solution. 11 plus 7 over 6, 18 over 6, 3, or the other one, 11 minus 7 over 6. That's 4, 6, otherwise known as 2 thirds. Now again, once I know the roots, first of all, I know that these are forbidden values. I cannot allow x to be 3 or 2 thirds, but I also know that x minus 3 and x minus 2 thirds are going to be factors of the denominator. So my denominator is going to look like x minus 3, x minus 2 thirds, and here I've made that note. I cannot let x be 3 or 2 thirds. Now, it's important to write this down because we're about to do something that's going to make that very critical. Now again, this gives us our variable factors. There may be an additional constant factor we need to complete the factorization. And in this case, you might note that if you expand this out, you get an x times an x, you get x squared. We really want a 3x squared, so we need a constant factor of 3 thrown in to complete our factorization. So there's our extra factor of a constant. Well, that numerator, we can do the same thing. We can find the roots of the equation x squared minus x minus 6 equal to 0, and I can solve this using the quadratic formula. Again, a equals 1, b equals negative 1, c equals negative 6. I drop those into the quadratic formula and I let the arithmetic dust settle. And after all the dust settles, I get my two roots, 6 over 2, 1 plus 5 over 2. The other root, 1 minus 5 over 2, negative 2. And the solutions are x equals 3, x equals negative 2. And the root theorem says solutions correspond to factors x minus 3, x minus negative 2. And so the numerator here factors as x minus 3 times x plus 2. And now I look for common factors in numerator and denominator. And because, and here's a very important thing to be careful with, because I have a product in the numerator and a product in the denominator, I am allowed to drop common factors. If you do not have a product, you cannot drop the common factor. So I do have a product in numerator, I have a product in the denominator, and so my common factor will drop out and I can cancel it out and leave my new expression this way, x plus 2 over 3 times x minus 2 thirds. And this is a perfectly good form to leave it in. I can combine this constant factor 3 with the denominator expression or not, your choice depends on what you want to do. So maybe we'll clean it up a little bit and so I'll multiply this here to get 3x minus 2. Now here's a very important thing to realize here. This prohibition, x cannot be 3, x cannot be 2 thirds, still holds. So this expression is this expression, as long as x is not equal to 3, not equal to 2 thirds. And the reason we still have to include this prohibition, these forbidden values, is that I can let x be 3 in this expression. If x is 3, there's no problems. If x is 3, there are problems on this. And I don't want no problem to be the same as big problem. So I include that prohibition. I still can't let x be 3. As long as x is not 3 or 2 thirds, these two expressions are equal. If x is 3, then these two expressions are definitely not equal. If x is 2 thirds, both of these expressions are meaningless.