Example: Divide ${3x^2 – x^3 – 3x + 5}$ by ${x – 1 – x^2}$, and verify the division algorithm.

Solution : Note that the given polynomials are not in standard form. To carry out division, we first write both the dividend and divisor in decreasing orders of their degrees.

So, dividend = ${–x^3 + 3x^2 – 3x + 5}$ and divisor = ${–x^2 + x – 1}$.

($-$)  ($+$)

  ($+$) ($-$)

     $3$

Now, Divisor × Quotient + Remainder = ${(–x^2 + x – 1) (x – 2) + 3}$ = ${–x^3 + x^2 – x + 2x^2 – 2x + 2 + 3}$ = ${–x^3 + 3x^2 – 3x + 5}$ = Dividend In this way, the division algorithm is verified.