Problem 5 · The value of x⁶ without complex numbers
Multiplying by $(x+1)$ to discover that $x^{3} = -1$.
Integer answer, at most 4 digitsIf $x^{2} - x + 1 = 0$, what is $x^{6}$? (Remark: this equation has no real solution, but you can find the value of $x^{6}$ without any knowledge of complex numbers.)
Copa Cangur · SCM
Easy
Closed answer
Reasoned solution
Key idea: $x^{2} - x + 1$ is precisely the factor in the sum of cubes: $(x+1)(x^{2}-x+1) = x^{3} + 1$.
Multiply the equation by $(x+1)$:
$$(x+1)(x^{2} - x + 1) = 0 \;\Longrightarrow\; x^{3} + 1 = 0 \;\Longrightarrow\; x^{3} = -1.$$
Therefore:
$$x^{6} = \left(x^{3}\right)^{2} = (-1)^{2} = 1.$$
Answer: 1