Question

The polynomials $(a{x}^3+3x^2-13)$ and $(2x^3-5x+a)$ are divided by $(x+2)$. If the remainder in each case is the same, find the value of $a$.

Answer 1

Let $p\left(x\right)=a{x}^3+3x^2-13$ and $g\left(x\right)=2x^3-4x+a$
By remainder theorem, the two remainders are $p\left(3\right)$ and $g\left(3\right)$
By the given condition, $p\left(3\right)=g\left(3\right)$
$\therefore p\left(3\right)=a.3^3+{3.3}^2-13=27a+27-13=27a+14$
$g\left(3\right)={2.3}^3-4.3+a=54-12+a=42+a$
Since $p\left(3\right)=g\left(3\right)$, we get $27a+14=42+a\therefore 26a=28$
$\therefore a=\frac{28}{26}=\frac{14}{13}$