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Algebra · Polynomial-Equations

Solve x^3 + 1 = 0

x^{3} + 1 = 0

Step by step

  1. x^{3} + 1 = 0

    Start from the equation as given.

  2. \left(x + 1\right) \left(x^{2} - x + 1\right) = 0

    Factor the polynomial (rational-root test + division).

  3. x + 1 = 0 \;\Rightarrow\; -1

    Set each factor equal to zero and solve it.

  4. x^{2} - x + 1 = 0 \;\Rightarrow\; \frac{1}{2} - \frac{\sqrt{3} \cdot i}{2},\ \frac{1}{2} + \frac{\sqrt{3} \cdot i}{2}

    Set each factor equal to zero and solve it.

  5. x = -1 ,\; x = \frac{1}{2} - \frac{\sqrt{3} i}{2} ,\; x = \frac{1}{2} + \frac{\sqrt{3} i}{2}

    All 3 roots.

Reveal the answer
x = -1 ,\; x = \frac{1}{2} - \frac{\sqrt{3} i}{2} ,\; x = \frac{1}{2} + \frac{\sqrt{3} i}{2}