Question

Find the relation between $m$ and $n$ if the polynomial $f\left(x\right)=2x^3+m{x}^2+nx-14$ has a factor $(x-1)$.

Answer 1

Given $f\left(x\right)=2x^3+m{x}^2+nx-14$.

If $f\left(x\right)$ has the factor $(x-1)$ then by remainder theorem we'll have the remainder will be $0$.

Then we'll have,

$f\left(1\right)=0$

or, $2\times 1^3+m\times 1^2+n\times 1-14=0$

or, $2+m+n-14=0$

or, $m+n=12$.

This is the required relation between $m$ and $n$.