Question

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

Answer 1

Step: Find the number

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

A polynomial f(x) has a factor $\text{(x-a)}$ if and only if $\text{f(a)}=0$.

$\text{f(x)}$ has a factor $\text{f(x-2)}$.
$\therefore \text{f(2)}=0$
$(2{)}^2+(2)+1-k=0$
$4+3-k=0$
$k=7$

Hence, $7$ should be subtracted from ${\text{x}}^2\text{+x+1}$.