Question

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

Answer 1

Given polynomial $x^3+a{x}^2+5$ and $x^3-2x^2+a$ divided by $x+2$ the remainder is same

Then $x+2=0$ or $x=-2$ replace $x$ by $-2$ we get

$p\left(x\right)=x^3+a{x}^2+5$

$p(-2)=(-2{)}^3+a(-2{)}^2+5$

$\Rightarrow p\left(2\right)=-8+a\times 4+5$

$\Rightarrow P\left(2\right)=4a-8+5$

$\Rightarrow p\left(2\right)=4a-3$

$q\left(x\right)=x^3-2x^2+a$

$\Rightarrow q(-2)=(-2{)}^3-2(-2{)}^2+a$

$\Rightarrow q\left(2\right)=-8-8+a$

$\Rightarrow A\left(2\right)=a-16$

$\therefore 4a-3=a-16$

Subtract a and add 3 both the sides we get,

$4a-3-a+3=a-16-a+3$

$\Rightarrow 3a=-13$

Divided both side by 3 we get

$a=-\frac{13}{3}$