Question

If both $(p+1)$ and $(p-1)$ are factors of $a{p}^3+p^2-2p+b$ find values of $a$ & $b$.

Answer 1

Given that,

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

$(p+1)$ and $(p-1)$ both are factor

If $p+1=0$ then $p=-1$ put the given equation.

$f(-1)=a{(-1)}^3+{(-1)}^2-2(-1)+b$

$f(-1)=-a+1+2+b$

$-a+b+3$

Then,

$-a+b-1=0$

$a-b+1=0$ …… (1)

If $p-1=0$ then $p=1$ put the given equation.

$\begin{align}

$f\left(1\right)=a{\left(1\right)}^3+{\left(1\right)}^2-2\left(1\right)+b$

$f(-1)=a+1-2+b$

$a+b-1$

Then,

$a+b-1=0$ …… (2)

On adding equation (1) and (2) to and we get,

$a=0$

Put $a=0$ in equation (1) and we get,

Then, $b=1$

Hence, $(0,1)$ is the answer.