Question

if x²+X+1=0 then what is the value of x²⁰¹⁵ + x²⁰¹³ + x​

Answer 1

Since $x^2 + x + 1 = 0$ means $x^2+x = -1$, we have for any $n$ the recursive relation

$x^n (x^2 + x) = x^{n+2} + x^{n+1} = -x^n \implies x^{n+2} = -x^{n+1} - x^n$

Let $f(x)=x^{2015}+x^{2013}+x$. Substitution using the recursive rule generates an alternating power-reduction pattern:

$f(x) = \left(-x^{2014} - x^{2013}\right) + x^{2013} + x = -x^{2014} + x$

$f(x) = -\left(-x^{2013} - x^{2012}\right) + x = x^{2013} + x^{2012} + x$

$f(x) = \left(-x^{2012} - x^{2011}\right) + x^{2012} + x = -x^{2011} + x$

$f(x) = -\left(-x^{2010} - x^{2009}\right) + x = x^{2010} + x^{2009} + x$

$f(x) = \left(-x^{2009} - x^{2008}\right) + x^{2009} + x = -x^{2008} + x$

$f(x) = -\left(-x^{2007} - x^{2006}\right) + x = x^{2007} + x^{2006} + x$

and so on.

Notice that in the equivalent forms of $f(x)$ involving 3 terms, the largest power of $x$ is a multiple of 3, and the next largest power is 1 less. This means after so many iterations of substitutions, we would end up with

$f(x) = x^3 + x^2 + x$

Then by factorizing, the expression of interest reduces to

$f(x) = x (x^2 + x + 1) = x \times 0 = \boxed{0}$