Question

Find the remainder when p(x) = 4x³ + 8x² − 17x + 10 is divided by (2x − 1).

Answer 1

Let:
$p\left(x\right)=4x^3+8x^2-17x+10$
$$\begin{gathered} \mathrm{And}, \\ 2x-1=0\Rightarrow x=\frac{1}{2} \end{gathered}$$
By the remainder theorem, we know that when $p\left(x\right)\text{is divided by}\left(2x-1\right),\text{the remainder is}p\left(\frac{1}{2}\right).$
Thus, we have:
$p\left(\frac{1}{2}\right)=4\times {\left(\frac{1}{2}\right)}^3+8\times {\left(\frac{1}{2}\right)}^2-17\times \frac{1}{2}+10$
$$\begin{gathered} =\frac{1}{2}+2-\frac{17}{2}+10 \\ =4 \end{gathered}$$

∴ Remainder = 4