Algebra • Step-by-Step

Solve x³ - x = 0 with Steps

Step-by-step cubic solution for x³ - x = 0. Factor out x to get x(x² - 1) = x(x - 1)(x + 1) = 0 yielding roots x = 0, 1, and -1.

Problem Statement

x^3 - x = 0

Final Result

x = -1, \; 0, \; 1

Formula Required: Common Factor & Difference of Squares

Factor out the greatest common factor x, then apply difference of squares.

x(x^2 - 1) = x(x - 1)(x + 1) = 0

Step-by-Step Solution

10 Steps

Write the equation in standard cubic form $ax^3 + bx^2 + cx + d = 0$.

x^3 - x = 0

Apply the Rational Root Theorem (a way to test for simple fraction/whole-number roots before trying harder methods). Testing $x = 0$, we find $f(0) = 0$, which proves that $(x - 0)$ is a factor by the Factor Theorem.

\begin{aligned} f(0) &= f(0) = (0)^3 - (0) = 0 + 0 = 0 \;\checkmark \\[4pt] &\implies (x - 0) \text{ is a factor} \end{aligned}

Perform polynomial division (long division / synthetic division) of $x^3 - x$ by the factor $(x - 0)$ to obtain the reduced quotient $x^2 - 1$. Both Long Division (NCERT standard) and Synthetic Division (shortcut) are shown below:

\begin{aligned} &\textbf{Method 1: Long Division (NCERT)} \\[4pt] &\begin{array}{r} x^2 - 1 \\ x - 0{\overline{\smash{\big)}\, x^3 - x}} \\ \underline{-(x^3 + 0x^2)\phantom{------------}} \\ 0x^2 - x\phantom{------} \\ \underline{-(0x^2 + 0x)\phantom{------}} \\ -x + 0\phantom{--} \\ \underline{-(-x + 0)\phantom{--}} \\ 0 \end{array} \\[10pt] &\textbf{Method 2: Synthetic Division (Shortcut)} \\[4pt] &\begin{array}{c|rrrr} 0 & 1 & 0 & -1 & 0 \\ & & 0 & 0 & 0 \\ \hline & 1 & 0 & -1 & 0 \end{array} \end{aligned}

Express the cubic equation in factored form and isolate the quadratic factor.

(x - 0)(x^2 - 1) = 0 \implies x^2 - 1 = 0

Calculate the discriminant $\Delta = b^2 - 4ac$ (tells us how many real roots exist) for $x^2 - 1 = 0$. Since $\Delta > 0$, there are two distinct real roots.

\Delta = (0)^2 - 4(1)(-1) = 0 - (-4) = 4

Apply the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$.

x = \frac{-(0) \pm \sqrt{4}}{2(1)} = \frac{0 \pm 2}{2}

Compute the two roots from the quadratic factor:

x_2 = \frac{0 + 2}{2} = 1, \quad x_3 = \frac{0 - 2}{2} = -1

Final summary: All three real roots of the cubic equation are:

x = 0, \; 1, \; -1

Combine into completely factored form and apply the Zero Factor Principle (if $a \cdot b = 0$, then $a = 0$ or $b = 0$).

x(x - 1)(x + 1) = 0

Structural Insight (Vieta's Relations for Cubics): Sum of roots = -b/a, Product of roots = -d/a.

\sum x_i = -\frac{b}{a} = 0, \quad \prod x_i = -\frac{d}{a} = 0

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Common Questions About Solve x³ - x = 0

What are the roots of x³ - x = 0?

The three roots are x = 0, x = 1, and x = -1.