Question

If $x^2+px+1$ is a factor of the expression $a{x}^3+bx+c$, then

• A. $a^2-c^2=ab$
• B. $a^2-c^2=-ab$
• C. $a^2+c^2=-ab$
• D. $a^2+c^2-ab=0$

Answer 1

The correct option is A

$a^2-c^2=ab$

Given that $x^2+px+1$ is factor of $a{x}^3+bx+c=0$, then

let $a{x}^3+bx+c=(x^2+px+1)(ax+\lambda )$, where λ is a costant. Then equating the coefficient of like powers of x on both sides, we get

$0=ap+\lambda ,b=p\lambda +a,c=\lambda$

⇒ $p=-\frac{\lambda }{a}=-\frac{c}{a}$

Hence b = $(-\frac{c}{a})c+a$
$\Rightarrow ab=a^2-c^2$