 You might as well accept it. You can run, but you can't hide. Fractions are inevitable. When we extend the notion of fractions to polynomials, we end up with what are known as rational expressions. A rational expression is a quotient of two polynomials. So in 1 over x, both numerator and denominator are polynomials, so this is a rational expression. Likewise, here we have a second-degree numerator and a first-degree denominator, so this is a rational expression. Here, our numerator is a constant, so it's a zero-degree polynomial, and our denominator is a first-degree polynomial, so this is a rational expression. Here, our numerator is a first-degree polynomial, but our denominator is a square root, and so this is not a rational expression. Because a rational expression is a quotient, and you can never divide by zero, it's possible for the expression to be undefined for certain values of the variable, and so this means you should always identify the values that make the expression undefined. Now, unless otherwise indicated, we'll focus on the real values that make the expression undefined, and we won't worry about complex numbers for now. For example, we might find what real values make our rational expression undefined. We can't let the denominator be zero, so solving denominator equal to zero. This is a quadratic equation, so we can use any method we want to to solve it, and we find x equals negative three, x equals negative one are solutions. Remember, if it's not written down, it didn't happen, and so we might make the observation since x equals negative three or x equals negative one make the denominator zero, we require x not be equal to negative three, and x not be equal to negative one. Pursuing the analogy with fractions, in a fraction we can cancel out common factors. If we have a times b over a times c, that common factor of a can be dropped, and we can get b over c. We can do almost the same thing with rational expressions. In a rational expression, we can simplify by cancelling common factors. x times y over x times z is y over z, provided we've already identified any forbidden values of the variable. So let's see if we can simplify this rational expression. So remember, the very first thing we should do any time we get a rational expression is to identify values that make the expression undefined. And so this means we have to see what makes the denominator zero and forbid those values. Since our denominator is x squared, we can't let it be zero, so solving denominator equals zero, which gives us x equals zero, and so since x equals zero makes the denominator zero, we require x not be equal to zero. Now our next step, well, since there's an x squared in the numerator and an x squared in the denominator, we can cancel them out and get 8x. And we can do this. Of course, this would be wrong. And the reason is that you can only cancel if numerator and denominator are products. So when we simplified, we had our numerator was the product a, b, or x times y. Our denominator was a product a, c, or x, z. And we can only simplify if we have a product in both numerator and denominator any other simplification is incorrect. In order to cancel, we must have a product in both numerator and denominator. Well, the denominator is a power which is automatically a product. The numerator is a sum, and so that means we need to factor the numerator. So a factor x squared plus 8x. And why not? It's helpful to write our denominator as a product and keeping in mind that we do want to cancel out common factors. We want to make sure that that product includes a factor of x. So instead of writing x squared, we can write this as x times x. And now we have a factor of x in the numerator, a factor of x in the denominator, and we can drop out that common factor. And so this allows us to simplify. And the important thing here is that we still have this original restriction, x cannot be equal to zero.