[MUSIC]. Let's know about the Remainder Theorem.

For example let's use the Remainder Theorem to find P(1), where P(x) = 2x^3 -

3x^2 - 9. Now the remainder theorem states the

following. if a polynomial P of x is divided by x

minus c then the remainder of that division is P evaluated at c.

Let's think about why this is true. By the division algorithm, when we divide

p(x) by x - c then p(x) is equal to x - C times some polynomial q(x) plus a

constant R. Where here Q is the quotient and R is the

remainder. And the reason this remainder is a

constant is because it has to be 1 less in degree than this divider here.

Which is of degree 1 so r would have to be of degree 0 or a constant.

Now let's evaluate P of C. That is P of C is equal to, we're putting

a C everywhere we see an X. So it's C minus C, times Q of C, plus R.

Or P of C. Is equal to 0 times Q of C plus R or sure

enough, P of C is this remainder R. Let's apply this to our problem.

We want to find P of 1. So our c here, is equal to 1.

So p of 1, will be the remainder, when we divide, p of x, by x minus c But let's

use long division here. Now x goes into 2x^3, 2x^2 * and 2x^2 * x

- 1. Is 2x cubed minus 2x squared.

When we subtract we get -x^2 - 9 and x goes into negative x^2 -x times and

negative x * x - 1 is -x^2 + x. And when we subtract here, we need to be

careful. Let's think of their being a +0x here.

So, when we subtract we have 0x - x or -x and then we still have the -9,

x goes into -x, -1 times. -1 * x - 1, is -x + 1.

And then when we subtract, we get -10 which is our remainder.

Therefore, p(1) is equal to this remainder, -10, which is our answer.

Now, we can check this answer by plugging this value, 1 into our polynomial, P.

Namely p(1) = 2 * 1^3 - 3 * 1^2 - 9 or 2 - 3 - 9, which sure enough is 2 - 12 or

-10. Let's look at another example.

Again let's use the remainder theorem to find p(-3) where p(x) = x^4 + 2x^3 - 4x^2

+ 5. Again in order to find p(c), we divide p

by x - c. Where here in this case, c = -3.

So, we would need to divide our polynomial by x - -3 or x + 3.

Again, in order to perform this division, we could either use long division or

synthetic division. Let's use synthetic division here.

So we write our c, -3 and then we put all the coefficients of P, namely 1, 2,

-4 and now be careful here. Remember, it's mandatory that we hold the

place of the x and write +0x. So we have a 0 and then our 5.

And then we drop the 1. -3 * 1 is -3.

We add and we get negative 1. And multiply we get 3, we add we get -1.

We multiply we get 3, we add we get 3. We multiply we get -9, when we add we get

-4. Now remember with synthetic division,

this last number here is our remainder. Therefore, that is, p(-3) by this

theorem. So our answer then is that p(-3) = -4.