Solve x⁴ + x³ - x² - x = 0
The solutions of x⁴ + x³ - x² - x = 0 are x = -1, -1, 0, 1 (repeated roots counted once each: -1, 0, 1).
How to work it out
- 1
A quartic has up to 4 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 0 (it is 0, so x = 0 is a root and x factors out).
- 2
Test the candidates until one gives 0. x = 0 is a root. Divide by (x − 0) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
x = 1 is a root. Divide by (x − 1) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 4
What is left is the quadratic x² + 2x + 1 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} + x^{3} - x^{2} - x = (x + 1)^{2}x(x - 1) = 0 \]
\[ x^{4} + x^{3} - x^{2} - x = 0 \]
\[ \begin{array}{c|ccccc} 0 & 1 & 1 & -1 & -1 & 0 \\ & 0 & 0 & 0 & 0 & 0 \\ \hline & 1 & 1 & -1 & -1 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 1 & 1 & 1 & -1 & -1 \\ & 0 & 1 & 2 & 1 \\ \hline & 1 & 2 & 1 & 0 \end{array} \]
\[ x^{2} + 2x + 1 = 0 \Rightarrow x = -1 \]
\[ x^{4} + x^{3} - x^{2} - x = (x + 1)^{2}x(x - 1) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 0 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -1 | -1 |
| Product of the roots | 0 | 0 |
Questions
What are the roots of x⁴ + x³ - x² - x = 0?
The solutions of x⁴ + x³ - x² - x = 0 are x = -1, -1, 0, 1 (repeated roots counted once each: -1, 0, 1).
How do you factor x⁴ + x³ - x² - x?
(x + 1)x(x − 1).
What is the sum of the roots?
-1 (it equals −1 by Vieta's formulas).