Question

If $(x-2)$ and $(x-3)$ are two factors of $x^3+ax+b$, then find the remainder when $x^3+ax+b$ is divided by $x-5$.

• A. $0$
• B. $15$
• C. $30$
• D. $60$

Answer 1

The correct option is D $60$

As $x-2$ and $x-3$ are factors of the given polynomial, $f\left(2\right)$ and $f\left(3\right)$ will be equal to $0$
So, $f\left(2\right)={\left(2\right)}^3+a\left(2\right)+b=0$
$=>8+2a+b=0$
$=>2a+b=-8$ --- (1)
And $f\left(3\right)={\left(3\right)}^3+a\left(3\right)+b=0$
$=>27+3a+b=0$
$=>3a+b=-27$ --- (2)
Solving eqns 1 and 2, we get $a=-19;b=30$
So, the polynomial is ${\left(x\right)}^3-19x+30$
And to get the remainder when it is divided by $(x-5)$, we need to calculate $f\left(5\right)$
So, $f\left(5\right)={\left(5\right)}^3-19\left(5\right)+30=125-95+30=60$