Question

Carry out the following divisions and write your answer in the form: Dividend = divisor × quotient + remainder.

(1) (x³ − 3x² + 6x − 27) ÷ (x − 3)
(2) (p⁴ − 12p² + 9) ÷ (p + 3)

Answer 1

(1) (x³ − 3x² + 6x − 27) ÷ (x − 3) can be shown as:
$$\begin{gathered} x^2+6 \\ x-3\overline{)x^3-3x^2+6x-27} \\ {x}^3-3x^2 \\ \underline{-+} \\ 0+0+6x-27 \\ 6x-18 \\ \underline{-+} \\ \underline{-9} \end{gathered}$$
$\therefore x^3-3x^2+6x-27=\left(x-3\right)\left(x^2+6\right)-9$

(2) (p⁴ − 12p² + 9) ÷ (p + 3) can be shown as:
$$\begin{gathered} p^3-3p^2-3p+9 \\ p+3\overline{)p^4-12p^2+9} \\ {p}^4+3p^3 \\ \underline{--} \\ -3p^3-12p^2+9 \\ -3p^3-9p^2 \\ \underline{++} \\ -3p^2+9 \\ -3p^2-9p \\ \underline{++} \\ 9p+9 \\ 9p+27 \\ \underline{--} \\ \underline{-18} \end{gathered}$$
$\therefore p^4-12p^2+9=\left(p+3\right)\left(p^3-3p^2-3p+9\right)-18$