Question

If the polynomials $2x^3+m{x}^2+3x-5$ and $x^3+x^2-4x+m$ leaves the same remainder when divided by $x-2$, then the value of $m$ is

• A. $-\frac{3}{13}$
• B. $-\frac{13}{3}$
• C. $\frac{3}{13}$
• D. $\frac{13}{3}$

Answer 1

The correct option is B $-\frac{13}{3}$

Let $p\left(x\right)=2x^3+m{x}^2+3x-5$ and $q\left(x\right)=x^3+x^2-4x+m$
By remainder theorem, the remainders are $p\left(2\right)$ and $q\left(2\right)$
But, $p\left(2\right)=q\left(2\right)$ $[$Given$]$
$\therefore 2{\left(2\right)}^3+m{\left(2\right)}^2+3\left(2\right)-5={\left(2\right)}^3+{\left(2\right)}^2-4\left(2\right)+m$
$\Rightarrow 16+4m+6-5=8+4-8+m$
$\Rightarrow 17+4m=4+m$
$\Rightarrow 3m=-13$
$\Rightarrow m=-\frac{13}{3}$