Answer :
To get rid of [tex] x^{3} [/tex], you have to take the third root of both sides:
[tex] \sqrt[3]{x^{3}} = \sqrt[3]{1} [/tex]
But that won't help you with understanding the problem. It is better to write [tex] x^{3}-1 [/tex] as a product of 2 polynomials:
[tex] x^{3}-1 = (x-1)\cdot (x^{2} +x +1) [/tex]
From this we know, that [tex] x-1 = 0 => x = 1 [/tex] is the solution. Another solutions (complex roots) are the roots of quadratic equation.
[tex] \sqrt[3]{x^{3}} = \sqrt[3]{1} [/tex]
But that won't help you with understanding the problem. It is better to write [tex] x^{3}-1 [/tex] as a product of 2 polynomials:
[tex] x^{3}-1 = (x-1)\cdot (x^{2} +x +1) [/tex]
From this we know, that [tex] x-1 = 0 => x = 1 [/tex] is the solution. Another solutions (complex roots) are the roots of quadratic equation.
[tex]x^3-1=0\\
(x-1)(x^2+x+1)=0\\
x-1=0\\
x=1\\
x^2+x+1=0\\
x^2+x+\frac{1}{4}+\frac{3}{4}=0\\
(x+\frac{1}{2})^2=-\frac{3}{4}\\
x+\frac{1}{2}=\sqrt{-\frac{3}{4}} \vee x+\frac{1}{2}=-\sqrt{-\frac{3}{4}}\\
x=-\frac{1}{2}+i\frac{\sqrt3}{2} \vee x=-\frac{1}{2}-i\frac{\sqrt3}{2}\\
x=-\frac{1-i\sqrt3}{2} \vee x=-\frac{1+i\sqrt3}{2}\\\\
x=\{1,-\frac{1-i\sqrt3}{2},-\frac{1+i\sqrt3}{2} \}[/tex]
