Question

Use synthetic division and the remainder theorem to determine $f\left(1/2\right)$ given that
$f\left(x\right)=-3x^5+x^3-2x^2+x-3$

Answer 1

From remainder theorem : $f\left(\frac{1}{2}\right)=$ Remainder of $\frac{f\left(x\right)}{x-\frac{1}{2}}$

So, let's divide $f\left(x\right)=-3x^5+x^3-2x^2+x-3$ by $\frac{1}{2}$ and we have

$$\begin{array}{cc}\begin{array}{c}\\ \frac{1}{2}\\ \end{array}& \begin{array}{cccccc}-3& 0& 1& -2& 1& -3\\ & -\frac{3}{2}& -\frac{3}{4}& \frac{1}{8}& -\frac{15}{16}& \frac{1}{32}\\ -3& -\frac{3}{2}& \frac{1}{4}& -\frac{15}{8}& \frac{1}{16}& -\frac{95}{32}\end{array}\end{array}$$

Hence, $f\left(\frac{1}{2}\right)=-\frac{95}{32}$