Sum of roots, product of roots

[crays]

Hi, roots problem again x(.
The roots of the equation x² +px + 1 = 0 are a and b. If one of the roots of the equation x² + qx + 1 = 0 is a³, prove that the other root is b³. [Done]

Without solving any equation, show that q = p(p² - 3). Obtain the quadratic equation with roots a⁹ and b⁹, giving the coefficients of x in terms of q.

Can't solve the last one, which is a⁹ and b⁹. I got
x² + (q³ -3p)x + 1.
it's suppose to be x^2 + [q²(q -3)]x + 1.

 

[dynamicsolo]

You don't mention it, but had you gotten the proof for $$q = p(p^2 - 3)$$ ? (It is pretty neat!)

I think you can just argue by analogy for the last proposition. $$a^9 = (a^3)^3$$, and similarly for $$b^9$$, so the linear coefficient -- call it 's' -- for that last quadratic equation ought to be

$$s = q(q^2 - 3)$$

(Is there a typo in your last line?)