Question

If yhe polynomials $3x^3+a{x}^2+3x+5$ and $4x^3+x^2-2x+a$ leave the the same remainder, when divided by $(x-2)$, then find the value of ${}^{\prime }{a}^{\prime }$ and the remainder in each case.

Answer 1

Let the given polynomials are $p\left(x\right)=3x^3+a{x}^2+3x+5$ and $f\left(x\right)=4x^3+x^2-2x+a$
According to question $p\left(2\right)=f\left(2\right)$
$3(2{)}^3+a(2{)}^2+3\left(2\right)+5=4(2{)}^3+(2{)}^2-2\left(2\right)+a$
$\Rightarrow 3\times 8+4a+6+5=32+4-4+a$
$\Rightarrow 24+4a+11=32+a$
$\Rightarrow 4a-a=32-35$
$\Rightarrow 3a=-3$
$\Rightarrow a=-1$
As the remainder is same, so $p\left(2\right)$ or $f\left(2\right)$ would be same when $a=-1$
$p\left(2\right)=3(2{)}^3+(-1\left)\right(2{)}^2+3\times 2+5[\therefore a=-1]$
$=24-4+6+5$
$=35-4=31$
Thus, $a=-1$ and $p\left(2\right)$ or $f\left(2\right)=31$