Well, notice in that second fraction, we can factor out a 2 in the numerator, and we get 2 times x minus 4. So we get a y squared over x minus 4 and we get a 2 x minus 8 over x minus 4.

We split them up into fractions with the common denominator. It's sort of the opposite of finding the common denominator. We can always split a fraction up by addition or subtraction in the numerator, so we can split this up into two fractions. Well, this one, we can't do any factoring because we have two different variables in the numerator, but here's something else we can do. We have a y squared plus 2x minus 8 over x minus 4. Two x squared time x minus 2, divided by x minus 7. We have an extra factor of x plus 2 that we can cancel.

The denominator is just an ordinary quadratic, so we factor that in the ordinary way. The expression on the top is a bit tricky, but notice that we can factor out a greatest common factor and then that just becomes a difference of two squares. And that is a simplified form of the same expression. We can cancel a factor of x minus 3 and we're left with x plus 7 over x minus 3. So when we factor, what we get is x minus 3 times x plus 7 in the numerator, x minus 3 squared in the denominator. In fact, the one in the denominator is just the square of a difference pattern. Well, we have an ordinary quadratic in the numerator and another ordinary quadratic in the denominator.

So, for example, suppose we're asked to simplify this rational expression. To simplify a rational expression, we have to factor the expressions in the numerator and the denominator, and then cancel the common factors. Many of the rational expressions you will see on the test will be ones that can be simplified, and your job will be to simplify them. So, for example, x minus 3 over x plus 2. Now, what exactly are rational expressions? A rational expression is a ratio, a fraction, of two algebraic expressions. In this lesson, we will examine rational expressions.

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