Solve x⁴ - 5x³ + 5x² + 5x - 6 = 0
The solutions of x⁴ - 5x³ + 5x² + 5x - 6 = 0 are x = -1, 1, 2, 3.
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 -6, so the candidates are ±1, ±2, ±3, ±6.
- 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.
- 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 - 5x + 6 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} - 5x^{3} + 5x^{2} + 5x - 6 = (x + 1)(x - 1)(x - 2)(x - 3) = 0 \]
\[ x^{4} - 5x^{3} + 5x^{2} + 5x - 6 = 0 \]
\[ \begin{array}{c|ccccc} 1 & 1 & -5 & 5 & 5 & -6 \\ & 0 & 1 & -4 & 1 & 6 \\ \hline & 1 & -4 & 1 & 6 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -1 & 1 & -4 & 1 & 6 \\ & 0 & -1 & 5 & -6 \\ \hline & 1 & -5 & 6 & 0 \end{array} \]
\[ x^{2} - 5x + 6 = 0 \Rightarrow x = 3,\ 2 \]
\[ x^{4} - 5x^{3} + 5x^{2} + 5x - 6 = (x + 1)(x - 1)(x - 2)(x - 3) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | 5 | 5 |
| Product of the roots | -6 | -6 |
Questions
What are the roots of x⁴ - 5x³ + 5x² + 5x - 6 = 0?
The solutions of x⁴ - 5x³ + 5x² + 5x - 6 = 0 are x = -1, 1, 2, 3.
How do you factor x⁴ - 5x³ + 5x² + 5x - 6?
(x + 1)(x − 1)(x − 2)(x − 3).
What is the sum of the roots?
5 (it equals −(-5) by Vieta's formulas).