Divide 3x^{3}
– 2x^{2} + 3x
– 4 by x
– 3 using synthetic division. Write
the answer in the form "
q(x) + ^{r(x)}/_{d(x)}
".

This question is asking
me, in effect, to convert an "improper" polynomial "fraction"
into a polynomial "mixed number". That is, I am being asked
to do something similar to converting the improper fraction
^{17}/_{5}
to the mixed number
3 ^{2}/_{5},
which is really the shorthand for the addition expression "3 + ^{2}/_{5}".

To convert the polynomial
division into the required "mixed number" format, I have to
do the division; I will show most of the steps.

First, write down
all the coefficients, and put the zero from x
– 3 = 0 (so
x
= 3) at the
left.

Next,
carry down the leading coefficient:

Multiply by the
potential zero, carry up to the next column, and add down:

Repeat
this process:

Repeat
this process again:

As you can see, the remainder
is 68.
Since I started with a polynomial of degree 3
and then divided by x
– 3 (that is, by
a polynomial of degree 1),
I am left with a polynomial of degree 2.
Then the bottom line represents the polynomial 3x^{2}
+ 7x + 24
with a remainder of 68.
Putting this result into the required "mixed number" format,
I get the answer as being:

It is always true that,
when you use synthetic division, your answer (in the bottom row) will
be of degree one less than what you'd started with, because you have divided
out a linear factor. That was how I knew that my answer, denoted by the
"3
7 24" in
the bottom row, stood for a quadratic, since I had started with a cubic.