Divide the following polynomials. Then place the answer in the proper location on the grid. Write your answer in order of descending powers of x. Write the quotient and remainder as a sum in this format: . Do not include parentheses in your answer. ( x^3 + y^3) ÷( x- y )

The answer is x²-y²  +  xy²+x²y/x-y
Solution:
By polynomial grid division, we start by the divisor (x - y) placed on the row headings of the table and end with the quotient on the column headings.
We know that x² must be in the top left of the grid so that the row and column multiply to x³. We multiply x² by the terms of the row headings to fill in all of the first column:
              x² 
      x      x³
     -y     -x²y

We got -x²y though we want y³. The next quadratic entry must then be -y² to get y³. Multiplying -y² by the divisor, we fill in all of the second column:
              x²       -y² 
      x      x³       -xy² 
     -y     -x²y      y³

We end up with a grid sum -x²y - xy² which tells us that we have a remainder of xy² + x²y that we write next to the grid:
              x²       -y² 
      x      x³       -xy²       remainder: + x²y + xy² 
     -y     -x²y      y³

We have to add the remaining fractional part to the quotient that we can read off the first row. Therefore,
     x³+y³ / x-y = x²-y²  +  xy²+x²y/x-y