Question

What number should be subtracted from $2{\text{x}}^3-5{\text{x}}^2+8\text{x}-17$ so that the resulting polynomial is exactly divisible by $(2\text{x}-5)$.​

Answer 1

Step: Find the number

Let's assume the required number to be k.
$\text{f(x)}=2{\text{x}}^3-5{\text{x}}^2+8\text{x}-17-\text{k}$

A polynomial $\text{f(x)}$ is exactly divisible by $\text{x-a}$ if and only if $\text{f(a)}=0$.

$\text{f(x)}$ is exactly divisible by $\text{(2x - 5)}$
$2\text{x}-5=0\Rightarrow \text{x}=\frac{5}{2}$
$\therefore \text{f}\left(\begin{array}{c}\frac{5}{2}\end{array}\right)=0$
$2{\left(\begin{array}{c}\frac{5}{2}\end{array}\right)}^3-5{\left(\begin{array}{c}\frac{5}{2}\end{array}\right)}^2+8\left(\begin{array}{c}\frac{5}{2}\end{array}\right)-17-\text{k}=0$
$2\left(\begin{array}{c}\frac{125}{8}\end{array}\right)-5\left(\begin{array}{c}\frac{25}{4}\end{array}\right)+\left(\begin{array}{c}\frac{40}{2}\end{array}\right)-17-\text{k}=0$
$\frac{125}{4}-\frac{125}{4}+20-17-\text{k}=0$

$3-\text{k}=0$
$\text{k}=3$

Hence, $3$ should be subtracted from $2{\text{k}}^3-5{\text{k}}^2+8\text{k}-17$.