If x^2+px+1 is a factor of the expression ax^ | vedclass

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(C) Given that $x^2+px+1$ is a factor of $ax^3+bx+c$,we can write:$ax^3+bx+c = (x^2+px+1)(ax+\lambda)$,where $\lambda$ is a constant.Expanding the rig

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$a^2-c^2=ab$

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(C) Given that $x^2+px+1$ is a factor of $ax^3+bx+c$,we can write:$ax^3+bx+c = (x^2+px+1)(ax+\lambda)$,where $\lambda$ is a constant.Expanding the right side:$ax^3+bx+c = ax^3 + (a p + \lambda)x^2 + (p\lambda + a)x + \lambda$.Comparing the coefficients of like powers of $x$ on both sides:Coefficient of $x^2$: $0 = ap + \lambda \implies p = -\frac{\lambda}{a}$.Constant term: $c = \lambda$.Substituting $\lambda = c$ into the expression for $p$,we get $p = -\frac{c}{a}$.Coefficient of $x$: $b = p\lambda + a$.Substituting $p = -\frac{c}{a}$ and $\lambda = c$ into the equation for $b$:$b = \left(-\frac{c}{a}\right)c + a$.$b = -\frac{c^2}{a} + a$.Multiplying by $a$:$ab = -c^2 + a^2$.$a^2 - c^2 = ab$.

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

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