Solve x³ + 3x² - 34x - 120 = 0
The solutions of x³ + 3x² - 34x - 120 = 0 are x = -5, -4, 6.
How to work it out
- 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 -120, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±20, ±24, ±30, ±40, ±60, ±120.
- 2
Test the candidates until one gives 0. x = -4 is a root. Divide by (x + 4) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
What is left is the quadratic x² - x - 30 = 0. Solve it (factoring or the quadratic formula):
- 4
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{3} + 3x^{2} - 34x - 120 = (x + 5)(x + 4)(x - 6) = 0 \]
\[ x^{3} + 3x^{2} - 34x - 120 = 0 \]
\[ \begin{array}{c|cccc} -4 & 1 & 3 & -34 & -120 \\ & 0 & -4 & 4 & 120 \\ \hline & 1 & -1 & -30 & 0 \end{array} \]
\[ x^{2} - x - 30 = 0 \Rightarrow x = 6,\ -5 \]
\[ x^{3} + 3x^{2} - 34x - 120 = (x + 5)(x + 4)(x - 6) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | -150 | no |
| -1 | -84 | no |
| 2 | -168 | no |
| -2 | -48 | no |
| 3 | -168 | no |
| -3 | -18 | no |
| 4 | -144 | no |
| -4 | 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 | 120 | 120 |
Questions
What are the roots of x³ + 3x² - 34x - 120 = 0?
The solutions of x³ + 3x² - 34x - 120 = 0 are x = -5, -4, 6.
How do you factor x³ + 3x² - 34x - 120?
(x + 5)(x + 4)(x − 6).
What is the sum of the roots?
-3 (it equals −3 by Vieta's formulas).