Solve x⁴ + 6x³ - x² - 54x - 72 = 0
The solutions of x⁴ + 6x³ - x² - 54x - 72 = 0 are x = -4, -3, -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 -72, so the candidates are ±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±18, ±24, ±36, ±72.
- 2
Test the candidates until one gives 0. x = -2 is a root. Divide by (x + 2) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
x = 3 is a root. Divide by (x − 3) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 4
What is left is the quadratic x² + 7x + 12 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} + 6x^{3} - x^{2} - 54x - 72 = (x + 4)(x + 3)(x + 2)(x - 3) = 0 \]
\[ x^{4} + 6x^{3} - x^{2} - 54x - 72 = 0 \]
\[ \begin{array}{c|ccccc} -2 & 1 & 6 & -1 & -54 & -72 \\ & 0 & -2 & -8 & 18 & 72 \\ \hline & 1 & 4 & -9 & -36 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 3 & 1 & 4 & -9 & -36 \\ & 0 & 3 & 21 & 36 \\ \hline & 1 & 7 & 12 & 0 \end{array} \]
\[ x^{2} + 7x + 12 = 0 \Rightarrow x = -3,\ -4 \]
\[ x^{4} + 6x^{3} - x^{2} - 54x - 72 = (x + 4)(x + 3)(x + 2)(x - 3) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | -120 | no |
| -1 | -24 | no |
| 2 | -120 | no |
| -2 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -6 | -6 |
| Product of the roots | -72 | -72 |
Questions
What are the roots of x⁴ + 6x³ - x² - 54x - 72 = 0?
The solutions of x⁴ + 6x³ - x² - 54x - 72 = 0 are x = -4, -3, -2, 3.
How do you factor x⁴ + 6x³ - x² - 54x - 72?
(x + 4)(x + 3)(x + 2)(x − 3).
What is the sum of the roots?
-6 (it equals −6 by Vieta's formulas).