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## Synthetic division of polynomials

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# Dividing polynomials: synthetic division

CCSS.Math:

## Video transcript

Let's do another synthetic
division example. And in another
video, we actually have the why this works relative
to algebraic long division. But here it's going
to be another just, let's go through
the process of it just so that you get
comfortable with it. And now is a good chance to
give it a shot, to actually try to simplify this
rational expression. So let's think about
this step by step. So the first thing
I want to do is write all of the coefficients
of the numerator. So I have a 2. Oh, I have to be careful here. Because the 2 is the
coefficient for x to the fifth, I have no x to the fourth term. Let me start over. So I have the 2 from
2x to the fifth. And then I have no
x to the fourth. So it's really 0x to the fourth. So I'll put a 0
as the coefficient for the x to the fourth term. And then I have a negative
1 times x to the third. And then I have a positive
3 times x squared. Negative 2 times x. And then I have a constant
term, or zero degree term, of 7. I just have a positive 7. And now let me just draw my
little funky synthetic division operator-looking symbol. And remember, the type of
synthetic division we're doing, it only applies when we are
dividing by an x plus or minus something. There's a slightly
different process you would have to do if it
was 3x or if was negative 1x or if it was 5x squared. This only works when we have
x plus or minus something. In this case we have x minus 3. So we have the negative 3 here. And the process
we show-- there's other ways of doing it-- is
you take the negative of this. So the negative of
negative 3 is positive 3. And now we're ready to perform
our synthetic division. So we'll bring down
this 2 and then multiply the 2 times the 3. 2 time 3 gives us 6. 0 plus 6 is 6. And then we multiply that times
the 3, and we get positive 18. Negative 1 plus 18 is 17. Multiply that times the 3. 17 times 3 is 51. 3 plus 51 is 54. Multiply that times 3. The numbers are getting
kind of large now. So that's going to be what? 50 times 3 is 150. 4 times 3 is 12. So this is going to be 162. Negative 2 plus 162 is 160. And then finally, 160
times 3 is going to be 480. And you add 480 to
7, and you get 487. And you can think of it, I only
have one term or one number to the left-hand side
of this bar here. Or I'm just doing the standard,
traditional x plus or minus something version of synthetic
division, I should say. So I can separate
this out, and now I've essentially gotten my answer. And it looks like voodoo,
and it kind of is voodoo. And that's why I
don't like to do it, because you're just
memorizing an algorithm. But there are other
videos why we explain why. And it can be fast
and convenient and paper saving very often,
like you see right here. But then we have
our final answer. It's going to be-- and
let me work backwards. So I'll start with
our remainder. So our remainder is 487. And it's going to be
487 over x minus 3. And so this is
our constant term. And so you're going to have plus
160 plus 487 over x minus 3. Now this is our x term. So it's going to be
54x plus all of this. This is going to be
our x squared term. So this is going to be 17x
squared plus 54x plus 160 and all of that. Then this is going to
be x to the third term. So this is going to be 6x to
the third plus all of that. And then finally, this is
our x to the fourth term-- 2x to the fourth. And let me erase this. So then I have my x
to the fourth term. So it is 2x to the fourth. And we are done. This thing simplifies
to this right over here. And I encourage you to verify it
with traditional algebraic long division.