Question

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

Answer 1

Given polynomial $p\left(x\right)=a{x}^3+3x^2-13$ and $2x^2-5x+a$ is divided by x-2 get same remainder

Then x-2=0 or x=2

$p\left(x\right)=a{x}^3+3x^2-13$

Replace x by 2 we get

$p\left(2\right)=a(2{)}^3+3(2{)}^2-13$

$\Rightarrow p\left(2\right)=8a+12-13$

$\Rightarrow p\left(2\right)=8a-1$

$2x^2-5x+a$

Replace x by 2 we get

$q\left(2\right)=2(2{)}^3-5(2)+a$

$\Rightarrow q\left(2\right)=16-10+a$

$\Rightarrow q\left(2\right)=6+a$

Given remainder is same

$\therefore 8a-1=6+a$

$\Rightarrow 8a-1+1-a=6+a+1-a$

$\Rightarrow 7a=7$

$a=1$