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Solve x³ - 3x - 2 = 0

The solutions of x³ - 3x - 2 = 0 are x = -1, -1, 2 (repeated roots counted once each: -1, 2).

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Graph of y = x³ - 3x - 2-2-1123-10-5510152025x = -1x = 2y
Graph of y = x³ - 3x - 2. The curve crosses the x-axis at x = -1, x = 2, the roots found above.

How to work it out

  1. 1

    A cubic has up to 3 real roots. The rational root theorem says that any integer root of a polynomial with integer coefficients and leading coefficient 1 divides the constant term -2, so the candidates are ±1, ±2.

    x3−3x−2=0x^{3} - 3x - 2 = 0
  2. 2

    Test the candidates until one gives 0. x = -1 is a root. Divide by (x + 1) with synthetic division; the bottom row is the quotient and a 0 remainder.

    −110−3−20−1121−1−20\begin{array}{c|cccc} -1 & 1 & 0 & -3 & -2 \\ & 0 & -1 & 1 & 2 \\ \hline & 1 & -1 & -2 & 0 \end{array}
  3. 3

    What is left is the quadratic x² - x - 2 = 0. Solve it (factoring or the quadratic formula):

    x2−x−2=0⇒x=2, −1x^{2} - x - 2 = 0 \Rightarrow x = 2,\ -1
  4. 4

    So the polynomial factors as

    x3−3x−2=(x+1)2(x−2)x^{3} - 3x - 2 = (x + 1)^{2}(x - 2)
LaTeX source of the formulas on this page
\[ x^{3} - 3x - 2 = (x + 1)^{2}(x - 2) = 0 \]
\[ x^{3} - 3x - 2 = 0 \]
\[ \begin{array}{c|cccc} -1 & 1 & 0 & -3 & -2 \\ & 0 & -1 & 1 & 2 \\ \hline & 1 & -1 & -2 & 0 \end{array} \]
\[ x^{2} - x - 2 = 0 \Rightarrow x = 2,\ -1 \]
\[ x^{3} - 3x - 2 = (x + 1)^{2}(x - 2) \]

Candidates tested (computed)

Candidate xp(x)Root?
1-4no
-10root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots00
Product of the roots22

Questions

What are the roots of x³ - 3x - 2 = 0?

The solutions of x³ - 3x - 2 = 0 are x = -1, -1, 2 (repeated roots counted once each: -1, 2).

How do you factor x³ - 3x - 2?

(x + 1)(x − 2).

What is the sum of the roots?

0 (it equals −0 by Vieta's formulas).

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    Solve x³ - 3x - 2 = 0: x = -1, 2 | WordGPT