Question

What number should be added to $2{\text{x}}^3-3{\text{x}}^2+7\text{x}-8$ so that the resulting polynomial is exactly divisible by $\text{(x - 1)}$.​

Answer 1

Step: Find the number

Let's assume the required number to be k.
$\text{f(x)=2}{\text{x}}^3\text{-3}{\text{x}}^2\text{+7}\text{x}\text{-8+}\text{k}$
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 $\text{x-1}$.

$\therefore \text{f(1)}=0$
$2\left(1\right)-3\left(1\right)²+7\left(1\right)-8+\text{k}$
$2-3+7-8+\text{k}=0$
$k=2$

Hence, $2$ should be added to $2{\text{x}}^3-3\text{x}²+7\text{x}-8$.