Hello

I've been looking at a question for some time, but haven't been able to find a way to start. The question is:

'If x^2 - px - q = 0, where p and q are positive integers, which of the following could not equal x^3?'

4x + 3, 8x + 5, 8x + 7, 10x + 3, 26x + 5

I know the equation has one positive and one negative root. I've completed the square and found the discriminant, but I can't find a way into the question. Could someone please give me a pointer.

Thank you.

I have a solution, but I'm not sure it's the neatest approach.

$x^2=px+q$

$x^3=px^2+qx$

But as $x^2=px+q$,

$x^3=p(px+q)+qx$

$x^3=p^2x+qx+pq$

$x^3=x(p^2+q)+pq$

We have these options:
$4x + 3$, $8x + 5$, $8x + 7$, $10x + 3$, $26x + 5$

First, I checked $26x+5$

p and q are integers and $pq=5$
This means $p$ and $q$ are $1$ and $5$, although I don't know which letter corresponds to which number.

Can I get $p^2+q=26$?
Yes! If $p=5$ and $q=1$! So I can rule this out. Check the others: I found one which didn't work.

Quacky, thank you for taking the time to reply. I see now how I should have approached the problem.