Question

Using remainder theorem, find the remainder when:
(81x⁴ + 54x³ − 9x² − 3x + 2) is divided by (3x + 2)

Answer 1

We have:
$3x+2=0\Rightarrow x=-\frac{2}{3}$
By the remainder theorem, we know that when f(x) is divided by (3x + 2), the remainder is f($-\frac{2}{3}$).
Thus, we have:
$$\begin{gathered} f\left(x\right)=\left(81x^4+54x^3-9x^2-3x+2\right) \\ \therefore f\left(-\frac{2}{3}\right)=\left[81\times {\left(-\frac{2}{3}\right)}^4+54\times {\left(-\frac{2}{3}\right)}^3-9\times {\left(-\frac{2}{3}\right)}^2-3\times \left(-\frac{2}{3}\right)+2\right] \\ =\left(16-16-4+2+2\right) \\ =0 \end{gathered}$$
​Hence, the required remainder is 0.