# factor theorm

• Jan 9th 2012, 05:23 PM
Ryan12
factor 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 PM
Quacky
Re: factor theorm
If $\displaystyle x-a$ is a factor, then $\displaystyle f(a)=0$
• Jan 9th 2012, 05:32 PM
Ryan12
Re: 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 PM
Quacky
Re: 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 PM
Ryan12
Re: factor theorm
then i simultaneously solve right???
thanks alot your help greatly appreciated,
• Jan 9th 2012, 06:05 PM
MarkFL
Re: 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.