Solve x⁴ + 3x³ - 6x² - 28x - 24 = 0
The solutions of x⁴ + 3x³ - 6x² - 28x - 24 = 0 are x = -2, -2, -2, 3 (repeated roots counted once each: -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 -24, so the candidates are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24.
- 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 = -2 is a root. Divide by (x + 2) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 4
What is left is the quadratic x² - x - 6 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} + 3x^{3} - 6x^{2} - 28x - 24 = (x + 2)^{3}(x - 3) = 0 \]
\[ x^{4} + 3x^{3} - 6x^{2} - 28x - 24 = 0 \]
\[ \begin{array}{c|ccccc} -2 & 1 & 3 & -6 & -28 & -24 \\ & 0 & -2 & -2 & 16 & 24 \\ \hline & 1 & 1 & -8 & -12 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -2 & 1 & 1 & -8 & -12 \\ & 0 & -2 & 2 & 12 \\ \hline & 1 & -1 & -6 & 0 \end{array} \]
\[ x^{2} - x - 6 = 0 \Rightarrow x = 3,\ -2 \]
\[ x^{4} + 3x^{3} - 6x^{2} - 28x - 24 = (x + 2)^{3}(x - 3) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | -54 | no |
| -1 | -4 | no |
| 2 | -64 | no |
| -2 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -3 | -3 |
| Product of the roots | -24 | -24 |
Questions
What are the roots of x⁴ + 3x³ - 6x² - 28x - 24 = 0?
The solutions of x⁴ + 3x³ - 6x² - 28x - 24 = 0 are x = -2, -2, -2, 3 (repeated roots counted once each: -2, 3).
How do you factor x⁴ + 3x³ - 6x² - 28x - 24?
(x + 2)(x − 3).
What is the sum of the roots?
-3 (it equals −3 by Vieta's formulas).