Question

Using factor theorem, show that:
(2x − 3) is a factor of (2x⁴ + x³ − 8x² − x + 6)

Answer 1

Let:
$f\left(x\right)=2x^4+x^3-8x^2-x+6$
Here,
$2x-3=0\Rightarrow x=\frac{3}{2}$
By the factor theorem, (2x $-$ 3) is a factor of the given polynomial if $f\left(\frac{3}{2}\right)=0$.
Thus, we have:

$$\begin{gathered} f\left(\frac{3}{2}\right)=\left[2\times {\left(\frac{3}{2}\right)}^4+{\left(\frac{3}{2}\right)}^3-8\times {\left(\frac{3}{2}\right)}^2-\left(\frac{3}{2}\right)+6\right] \\ =\left(\frac{81}{8}+\frac{27}{8}-18-\frac{3}{2}+6\right) \\ =0 \end{gathered}$$
Hence, (2x $-$ 3) is a factor of the given polynomial.