Given that x-1 and x+2 are factors of f(x)= x^3 + px + q where p and q are integers, find p and q.

How do i start in find p and q????

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- Jan 9th 2012, 05:23 PMRyan12factor theorm
Given that x-1 and x+2 are factors of f(x)= x^3 + px + q where p and q are integers, find p and q.

How do i start in find p and q???? - Jan 9th 2012, 05:25 PMQuackyRe: factor theorm
If $\displaystyle x-a$ is a factor, then $\displaystyle f(a)=0$

- Jan 9th 2012, 05:32 PMRyan12Re: factor theorm
so wait if i use x-1 then where x is, it would be replaced by 1,

F(1) = 1^3 + p(1) + q

and i was just end up where i started, so how to advance on this questions , and thanks - Jan 9th 2012, 05:37 PMQuackyRe: factor theorm
$\displaystyle F(1) = 1^3 + p(1) + q = 0$

This gives you one equation in $\displaystyle p$ and $\displaystyle q$. Now find another using the other factor. - Jan 9th 2012, 06:02 PMRyan12Re: factor theorm
then i simultaneously solve right???

thanks alot your help greatly appreciated, - Jan 9th 2012, 06:05 PMMarkFLRe: factor theorm
Another method:

$\displaystyle x^3+px+q=(x-1)(x+2)(x-k)$

We know k is real as complex roots come in conjugate pairs.

$\displaystyle x^3+px+q=\left(x^2+x-2\right)(x-k)$

$\displaystyle x^3+px+q=\left(x^3+x^2-2x\right)-\left(kx^2+kx-2k\right)$

$\displaystyle x^3+0\cdot x^2+px+q=x^3+(1-k)x^2-(2+k)x+2k$

Now equate coefficients.