0:00

[MUSIC] Let's learn about synthetic division.

[SOUND] For example, let's use synthetic division to divide 2x^3 - 9x^2 + 6x - 7

by x - 4. Now, synthetic division is a shortcut

used to divide a polynomial by x - r. Notice here, this is a degree one

polynomial. So, synthetic division can only be used when the divisor is a degree

one polynomial. And this technique comes from looking at

the coefficients of P(x) and the value of r.

Now, what are the coefficients of P(x) here? Well, let's put them under a

division sign. We have 2, -9, 6 and -7.

And it is imperative that our polynomial is in standard form before we read off

these coefficients. But moreover if there were any missing

powers of x here, we'd have to write 0 for that power as a place holder.

Now, what is our r here? Our r is 4. So let's put the 4 to the left of the

division symbol. Now, that these numbers are set up, the

1st step in synthetic division is to drop the first coefficient of P so we'll drop

the 2. The next step is that we multiply the 4

and the 2, and put our result here. So, 4 * 2 is 8.

And then, we add. So, -9 + 8 = -1. And now, we continue this pattern.

So, we multipy the 4 and the -.1, which gives us -4.

Again, we're going to add, which gives us 2 and then we multiply, which gives us 8

and then we add, which gives us 1. Now, what are these numbers down here?

These first three numbers are the coefficients of the quotient of the

division. And this last number here is the

remainder of the division. And moreover, the degree of this quotient

is 1 less than the degree of our dividend, P(x).

And this is because our divisor x - r is a degree one polynomial.

And since our dividend here, P(x), is a degree three polynomial,

then our quotient will be a degree two polynomial.

Therefore, are quotient, which we'll call Q(x), is equal to a degree two polynomial

with these coefficients. Namely, it's 2x^2 - 1 * x + 2.

And our remainder is equal to 1, this number here, this last number.

Therefore, by the division algorithm P(x) / x -r is equal to Q(x) which is 2x^2 - x

+ 2 plus remainder divided by the divisor which is x - 4, in this case.

Let's actually take a quick look at the long division so you can see why this

technique works. So, we'll be dividing P(x) by x - 4.

Now, x goes into 2x^3, 2x^2 timesm and 2x^2 * x - 4 is 2x^3 - 8x^2.

And now subtracting, we get -x^2 + 6x - 7 and x goes into -x^2

-x times, And -x * x - 4 is -x^2 + 4x.

And subtracting, gives us 2x - 7. And now, x goes into 2x, two times, and 2

* x - 4 is 2x - 8, and when we subtract, we get 1.

So, sure enough, our quotient is 2x^2 - x + 2, and our reminder down here is 1.

Now, sometimes students get confused that we subtract here but we add with

synthetic division. But notice here, we have this negative in

front of the 4. Whereas, with synthetic division, we don't.

So, subtracting then here will give us the same result as adding when we use

synthetic division. [SOUND] Alright, let's look at 1 more

example. Let's use synthetic division to divide

P(x) by x + 4. Now, the first thing to notice about P(x)

here is that it's not written in standard form.

So, let's do that, let's write it in standard form.

We have P(x) is equal to, we have this x^4 term, so -2x^4, and then

we have this x^3 term, so -7x^3. And then, we have this x^2 term, so plus

4x^2, and finally, we have this constant, +1.

Now, notice here that there is no x term. And with synthetic division it's very

important that we have every power of x. That is, before reading off the

coefficients of P, we need to write P(x) as follows, -2x^4 - 7x^3 + 4x^2 + 0x.

We have to have a place holder and then, plus 1.

Now, we're ready to read of the coefficients of P.

They're -2, -7, 4, 0, and 1. But what is r? We're dividing P by x + 4,

and we can think of x + 4 as x minus a -4.

So, this is r here. So, let's put the -4 here.

And now, we drop, we multiply, which gives us 8.

Then, we add, which gives us 1, then we multiply the -4 and the 1, which gives us

-4, and then we add which gives us 0, and then -4 * 0 is 0.

We add, we get 0 again, and -4 * 0 = 0. We add, we get 1.

Now remember, these are the coefficients of our quotient.

But moreover, our quotient is 1 less in degree than this dividend here, P(x),

which is of degree 4. Which means Q(x) then, our quotient,

is of degree three with these coefficients, therefore, it is equal to

-2x^3 + 1x^2 + 0x + 0 which we don't need to write.

And this is our remainder, r(x). Therefore, by the division algorithm, we

have that P(x) / x - r is equal to the quotient Q(x) plus the remainder r(x) / x

- r. Or in our case,

P(x) is 1 + 4x^2 - 2x^4 - 7x^3 / x - r or x + 4,

is equal to Q(x), which is -2x^3 + x^2 plus the remainder of 1 / x + 4.

Alright, and this is how we can use synthetic division to divide a polynomial

by a divisor that is a degree one polynomial.

Thank you and we'll see you next time. [MUSIC]