If x^2 + px + 1 is a factor of ax^3 + bx + c, | vedclass

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(C) Since $x^2 + px + 1$ is a factor of $ax^3 + bx + c$,we can write:$ax^3 + bx + c = (x^2 + px + 1)(ax + c)$Expanding the right side:$ax^3 + bx + c =

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$a^2 - c^2 = ab$

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(C) Since $x^2 + px + 1$ is a factor of $ax^3 + bx + c$,we can write:$ax^3 + bx + c = (x^2 + px + 1)(ax + c)$Expanding the right side:$ax^3 + bx + c = ax^3 + cx^2 + pax^2 + pcx + ax + c$$ax^3 + bx + c = ax^3 + (c + ap)x^2 + (pc + a)x + c$Comparing the coefficients of $x^2$ and $x$:$1) \, c + ap = 0 \Rightarrow p = -c/a$$2) \, pc + a = b$Substitute $p = -c/a$ into the second equation:$(-c/a)c + a = b$$-c^2/a + a = b$$-c^2 + a^2 = ab$$a^2 - c^2 = ab$

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

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