Question

If $x-2$ is a factor of each of the following two polynomials, find the value of $a$ in each case

$x^3-2a{x}^2+ax-1$

Answer 1

Given: $(x-2)$ is a factor of $x^3-2a{x}^2+ax-1$.

Let, $f\left(x\right)=x^3-2a{x}^2+ax-1$

$\because (x-2)$ is a factor of $x^3-2a{x}^2+ax-1$

$\therefore f\left(2\right)=0$ [ Using Factor Theorem]

$\Rightarrow (2{)}^3-2a\times (2{)}^2+a\times \left(2\right)-1=0$

$\Rightarrow 8-8a+2a-1=0$

$\Rightarrow -6a+7=0$

$\Rightarrow -6a=-7$

$\therefore a=\frac{7}{6}$

Hence, the value of $a=\frac{7}{6}$.