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:$ax^3+bx+c = (x^2+px+1)(ax+k)$ for some constant $k$.Expanding the right side:$ax^3+

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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:$ax^3+bx+c = (x^2+px+1)(ax+k)$ for some constant $k$.Expanding the right side:$ax^3+bx+c = ax^3 + (ap+k)x^2 + (p k+a)x + k$.Comparing the coefficients of $x^2$,$x$,and the constant term on both sides:$1$) Coefficient of $x^2$: $ap+k = 0 \implies k = -ap$.$2$) Constant term: $k = c$.$3$) Coefficient of $x$: $pk+a = b$.From $(1)$ and $(2)$,we have $c = -ap$,so $p = -c/a$.Substitute $p = -c/a$ and $k = c$ into the equation for the coefficient of $x$:$(-c/a)(c) + a = b$$-c^2/a + a = b$Multiply by $a$:$-c^2 + a^2 = ab$$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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