Question

If $x^2-1$ is a factor of $a{x}^2+b{x}^3+c{x}^2+dx+e$, then

• A. $a{x}^2+cx+e$ divisible by $(x-1)$
• B. $a{x}^2+cx+e$ divisible by $(x+1)$
• C. $b{x}^2+d$ divisible by $(x-2)$
• D. None of these

Answer 1

The correct option is A $a{x}^2+cx+e$ divisible by $(x-1)$

$p\left(x\right)=a{x}^2+b{x}^3+c{x}^2+dx+e$

$p\left(1\right)=a+b+c+d+e=0$ as $(x-1)$ is root

$p(-1)=a-b+c-d+e=0$ as $(x+1)$ is root

$a+c+e=b+d=0$

in option (A),$g\left(x\right)=a{x}^2+cx+e$

$g\left(1\right)=a+c+e=0$ so g(x) is divisible by $(x-1)$