If x^2+p x+1 is a factor of a x^3+b x+c,then | vedclass

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(C) Given that $x^2+p x+1$ is a factor of $a x^3+b x+c$. Let $a x^3+b x+c = (x^2+p x+1)(a x+\alpha)$. Expanding the right side: $a x^3+b x+c = a x^3 +

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$a^2-c^2=a b$

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(C) Given that $x^2+p x+1$ is a factor of $a x^3+b x+c$. Let $a x^3+b x+c = (x^2+p x+1)(a x+\alpha)$. Expanding the right side: $a x^3+b x+c = a x^3 + (p a+\alpha) x^2 + (p \alpha+a) x + \alpha$. Comparing the coefficients of $x^2, x^1,$ and the constant term: $1) \ p a + \alpha = 0 \implies p = -\frac{\alpha}{a}$ $2) \ p \alpha + a = b$ $3) \ \alpha = c$ Substituting $\alpha = c$ into the first equation: $p = -\frac{c}{a}$. Now,substitute $p = -\frac{c}{a}$ and $\alpha = c$ into the second equation: $(-\frac{c}{a})(c) + a = b$ $-\frac{c^2}{a} + a = b$ $-c^2 + a^2 = a b$ $a^2 - c^2 = a b$.

If $x^2+p x+1$ is a factor of $a x^3+b x+c$,then

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