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I came across this exam question:

$f(x)=x^3+5x^2+px-q$

Given that $(x+2)$ and $(x-1)$ are factors of $f(x)$,

a) form a pair of simultaneous equations in $p$ and $q$

b) show that $p=2$ and find the value of $q$

I immediately wrote, $f(x)=(x+2)(x-1)(x+\alpha) = x^3+5x^2+px-q$, expanded it and equated the coefficients. This answers the second part of the question directly, and given $p$ and $q$ I can construct simultaneous equations involving them arbitrarily.

What rule does the a) part of the question expect me use?

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    $\begingroup$ You can note that $f(-2)=f(1)=0$. $\endgroup$ – SMM Sep 11 '18 at 9:04
  • $\begingroup$ For the rule's name, it's called Factor Theorem. $\endgroup$ – Theo Bendit Sep 11 '18 at 9:17
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Just plug in the roots $x=-2$ and $x=1$ to get $$ 0=12-2p-q \\ 0=6+p-q. $$

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