If {x^2} + px + 1 is a factor of the expressi | vedclass

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(C) Given that ${x^2} + px + 1$ is a factor of $a{x^3} + bx + c$,we can write:$a{x^3} + bx + c = (x^2 + px + 1)(ax + k)$where $k$ is a constant.Expand

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${a^2} - {c^2} = ab$

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

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

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