Solving Polynomial Equations with Exponents of 40

[heartyface]

Polynomials help~~

Heh, so I posted this thread in the wrong category so I'm reposting it! =)
Hello. So here was this problem I came across:

If x^4-x^3+x^2-x^1+x^0=0, what is the numerical value of x^40-x^30+x^20-x^10+x^0?
I did try doing many stuffs (symmetry) & factoring, but I think none of these steps helped.
Enlighten the youngster, gracias.

 

[adaptation]

If 2X=0, what is 20X?

 

[heartyface]

@adaptation -_- very funny, if it was a simple problem like that I wouldn't have posted it.
Btw, the answer's 1.
Hmm... but How...?

 

[kushan]

I found no shortcut in doing this :uhh: ,
maybe then the only way to do this is find the roots and put them in the second equation :grumpy: ,
Yea lengthy it is

 

[kushan]

There could be a work around , using roots of unity , you need to think about it .
( sum of roots of unity and their properties )

 

[Deveno]

our original polynomial is p(x) = x⁴-x³+x²-x+1.

note that p(-1) = 1 + 1 + 1 + 1 + 1 = 5, so -1 is not a root of p(x).

now consider q(x) = (x+1)p(x) = x⁵ + 1.

Mod note: rest of solution removed