Hi, Henriettas.

If I may add to Ronald's example ...

If the volume did happen to be (8x^3 + 8x^2 - 8x - 5), and the area happened to be (4x^2 + 2x - 5), we could solve for the height by factoring.  But if you're looking for directions for polynomial long division, your problem might look something like this:

                                                                                                                    .
4x^2  + 2x  - 5 ) 8x^3 + 8x^2 - 8x - 5    

To divide these polynomials, start by looking at how many times 4x^2 "goes into" 8x^3.  You can quickly figure that out by writing 8x^3 over 4x^2 and reducing:

8x^3
4x^2

This is 2x.  Now place the 2x in the quotient:

                     ____               2x                                                       .
4x^2 + 2x - 5 ) 8x^3 + 8x^2 - 8x - 5   

Now multiply the 2x by all three terms of the divisor (4x^2 + 2x - 5) and write them under the dividend (8x^3 + 8x^2 - 8x -5), and be sure to subtract:

                     ______             2x                                                          . 
4x^2 + 2x - 5 ) 8x^3 + 8x^2 - 8x - 5
                   - ( 8x^3 + 4x^2 -10x    )                                                        .

                     ___                  2x                                                             .
4x^2 + 2x - 5 ) 8x^3 + 8x^2 - 8x - 5  
                    -  8x^3 - 4x^2  +10x                                                            .
                                4x^2   + 2x - 5

Now, do this division process again... 

                     _____              2x + 1
4x^2 + 2x - 5 ) 8x^3 + 8x^2 - 8x - 5
                     - 8x^3 - 4x^2 +10x                                                        .
                                 4x^2 + 2x - 5
                              - (4x^2 + 2x - 5)    

                      ________        2x + 1
4x^2 + 2x - 5 ) 8x^3 + 8x^2 - 8x - 5
                     - 8x^3 - 4x^2 +10x                                                  .
                                  4x^2 + 2x - 5
                                - 4x^2 - 2x + 5       
                                                   0  

Since there is no remainder, the answer to Ronald's problem is again 2x + 1 

Hope this helps with polynomial long division, Henriettas.
        

well, Volume is the area of the rectagle times the height...so if you have the area expressed as a polynomial and the volume expressed as a polynomial all you need to do is factor the volume polynomial and divide out the area polynomial.  Since you know the area has to be one of the factors the process should be made slightly easier:

Say the volume is given as (8x^3 + 8x^2 - 8x - 5) and the area is given as (4x^2 + 2x - 5).  If we simplify the volume with the area as a known factor we get (4x^2 + 2x -5)(2x + 1)=0.  If we divide out the area polynomial the (4x^2 + 2x - 5) cancels out and becomes 1 and the answer for the height is (2x + 1).

Hope this helps. Ron M. If anyone sees anything wrong with this answer let me know