Question

If the polynomials $2x^3+a{x}^2+4x-12$ and $x^3+x^2-2x+a$ leave the same remainder when divided by (x-3), find the value of a. Also find the remainder.

Answer 1

Let $p\left(x\right)=2x^3+a{x}^2+4x-12$,
$q\left(x\right)=x^3+x^2-2x+a$.
From polynomial remainder theorem, When p(x) is divided by (x-3) then the remainder is p(3).
Now, $p\left(3\right)=2(3{)}^3+a(3{)}^2+4\left(3\right)-12$
$=2\left(27\right)+a\left(9\right)+12-12$
$=54+9a$ ....(1)
When q(x) is divided by $(x-3)$ the remainder is q(3).
Now, $q\left(3\right)=2(3{)}^3+(3{)}^2+2\left(3\right)+a$
$=\left(27\right)+\left(9\right)-6+a$
$=30+a$ ......(2)
Given that p(3)=q(3).
That is, $54+9a=30+a$ (By (1) and (2))
$\Rightarrow 9a-a=30-54$
$\Rightarrow 8a=-24$
$\therefore a=-\frac{24}{8}=-3$
Substituting a=-3 in p(3), we get
$p\left(3\right)=54+9(-3)=54-27=27$
$\therefore$ The remainder is 27.