What you want to do is polynomial division.  Basically the overall view is that you teach each component (ax^n) as a different digit and the rest follows like normal division.  Follow the steps in this link, and feel free to ask questions if you don't understand part of it.

http://www.sosmath.com/algebra/factor/fac01/fac01.html

We have to show our work using long division. No whole number times 2 gives you 3. Do you just use a fraction down in the long division work? This is where I am confused.

-- How do I show this below the problem using long division?

Hi JaneFairfax!

That's a very interesting approach.  I'll remember that.

And Xerxes,  polynomial division works pretty much like division in arithmetic.  Let's see if I can get
the numbers to line up fairly nicely in a post.  I'll just use the coefficients.

The 1 of the divisor 1  1  2 is the estimator for the process.

                      3  2  -1             Estimate 1 into 3 to get the 3
                   _____________
       1  1  2  |   3  5  7  3  -2     
                      3  3  6              multiply the 3 times 1  1  2 to get this line
                      ______
                           2  1  3       Subtract the 3  3  6 from the 3  5  7 to get 2  1 and bring down the 3
                           2  2  4       Estimate 1 into 2 and multiply 1  1  2 by 2 to get 2  2  4
                           ______         
                              -1  -1  -2    Subtract 2  2  4 from 2  1  3 to get - 1  -1 and bring down the -2
                              -1  -1  -2    Estimate 1 into -1 and multiply 1  1  2 by -1 to get -1  -1  -2
                              ________
                                         0    Subtract to get the remainder of zero.

If the remainder line had been something like 2  -3  then this would be 2x-3 in long form.

The basic algorithm is "estimate (division), then multiply, the subtract just like in arithmetic,
except now we are dealing with signed numbers.  If you get double digit numbers in the process
you cannot carry between columns line in arithmetic.

This is a "repeated subtraction" algorithm and can be done in a short (place value) form since
the polynomials are place value in an unknown base x.  But the process is basically the same
as that done in arithmetic except for now dealing with signed numbers.

And by the way, if you do get an estimation like 3 into 2 in a polynomial division then the number
to write up top is the fraction 2/3.  This makes the following arithmetic quite messy most of the
time, but we have no choice since we can't "borrow" in unknown base arithmetic.

Most authors of algebra books avoid these nasty problems like the plague.  They are quite happy
for the most part to "rig" the problems to be nice.  It makes no difference to a program like
Maple whether the numbers "turn out nice" or not.  It just crunches it out.  But to us humans it
makes a big difference.

noelevans wrote:

Hang in there!  You'll get it.

I hope he gets it by now.  This is a 5 year old thread.

Hi;

You can answer posts that are old if you feel you have something new to add.

Hi noelevans;

Morning to you! It would be a funny looking sweater!

Hi JFF;

My Jana, wunderbar! A clever idea and you correctly guessed that the school problem did not have a remainder.

But if it had been like the above problem that method of equating the coefficients gets a little bit nasty. This is due to the fact that your idea will tend to balance the remainder in b and c. This may not be desirable.