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# Roots of Polynomial equations (FP1) watch

1. How do i find the roots of this eqn?

Q: Find the eqn whose roots are the fourth powers of the roots of the eqn x^3 + x + 1 = 0

I understand the question, just cant see how to get the roots
2. You don't need to find the roots -- and in fact two of the roots are complex

Call the roots of that equation and .

So

You need to use combinations of these to find and and (the last one is easy enough).

Then you can form the equation.
3. lots of ways,
substitute
and choose m so that you get only terms in w^3, w^0 and w^(-3)?

edit: take Glutamic acid's answer, I was purely trying to find the roots;
4. I think a substitution method might be quicker than Glutamic Acid's method.

Rearrange and do some squaring and you get the answer pretty quickly.
5. I have never seen the substitution method explained to my satisfaction in any book (probably because I'm a bit simple )..so I'm going to try here with this as an example.

Spoiler:
Show

If are the roots of

then

and if then

and so

Rearranging ...

Squaring

Rearranging

Squaring

and rearranging again

and so a cubic with the required roots is

I'm sure this isn't perfect and someone out there can do better. If so please do.

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Updated: November 10, 2008
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