[MUSIC] Let's look at simplifying a ratio of polynomials.

[SOUND] For example, let's simplify this expression here.

The first thing we can do is factor a 3 out of the numerator, which gives us 3 *

x^2 + x - 6 all divided by x^2 + 4x + 3. And now, we can factor both quadratic

expressions in the numerator and denominator, which gives us 3 * x + 3 * x

- 2 / x + 3 * x + 1. And then, we can cancel these common

factors of x + 3 and we're assuming here, of course, that x does not equal -3,

which leaves us with our answer of 3 * x - 2 / x + 1.

Alright, let's look at another example. [SOUND] Let's simplify this expression

here. Again, we can begin by factoring out the

common factor in the numerator which is 4 here.

which leaves us with 1 - w^2 still divided by w^2 + w - 2.

And now, we can factor about the numerator and denominator.

The numerator here is the difference of two squares and it factors into 1 - w * 1

+ w. And the denominator will factor into w -

1 * w + 2. Now, although these factors are not

exactly the same, they're the negatives of each other, aren't they? That is, w -

1 = 1 - w. And to see this, we can distribute the

negative or -1 through to both of these terms,

which would give us -1 + w which is w - 1.

So, let's use this fact here. That is, this is equal to 4 * 1 - w * 1 +

w and now, divided by -1 - w * w +2. And now, we can cancel these common

factors of 1 - w and we're assuming here, of course, w does not equal 1, which

leaves us with our answer of -4 * 1 + w / w + 2.