Question

The table represents the start of the division of 3x^4+22x^3+37x^2-7x+10by the indicated divisor

Answer 1

The quotient is x^3 + 4x^2 -x + 1.

Solution:
By polynomial grid division, we start by the divisor 3x + 10 placed on the column headings.
               3x      10
     x^3    3x^4   

We know that 3x^4 must be in the top left which means that the first row entry must be x^3. So the row and column multiply to 3x^4. We use this to fill in all of the first row, multiplying x^3 by the terms of the column entries.
                 3x       10
     x^3      3x^4   10x^3 
     4x^2

We now got 10x^3 though we want 22x^3. The next cubic entry must then be 12x^3 so that the overall sum is 22x^3.
                  3x         10
     x^3      3x^4     10x^3 
     4x^2    12x^3

Now we have 40x^2, so the next quadratic entry must be -3x^2 so that the overall sum is 37x^2.
                 3x          10
     x^3      3x^4      10x^3 
     4x^2    12x^3    40x^2
     -x       -3x^2      -10x

This time we have -10x, so the next linear entry must be 3x so that the overall sum is 7x.
                 3x          10
     x^3      3x^4     10x^3 
     4x^2    12x^3    40x^2
     -x         -3x^2    -10x
     1           3x         10

The bottom and final term is 10, which is our desired answer. Therefore, we can now read the quotient off the first column:
     3x^4+22x^3+37x^2-7x+10 / 3x + 10 = x^3 + 4x^2 -x + 1