Solve x³ + 15x² + 74x + 120 = 0
The solutions of x³ + 15x² + 74x + 120 = 0 are x = -6, -5, -4.
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 + 11x + 30 = 0. Solve it (factoring or the quadratic formula):
- 4
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{3} + 15x^{2} + 74x + 120 = (x + 6)(x + 5)(x + 4) = 0 \]
\[ x^{3} + 15x^{2} + 74x + 120 = 0 \]
\[ \begin{array}{c|cccc} -4 & 1 & 15 & 74 & 120 \\ & 0 & -4 & -44 & -120 \\ \hline & 1 & 11 & 30 & 0 \end{array} \]
\[ x^{2} + 11x + 30 = 0 \Rightarrow x = -5,\ -6 \]
\[ x^{3} + 15x^{2} + 74x + 120 = (x + 6)(x + 5)(x + 4) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 210 | no |
| -1 | 60 | no |
| 2 | 336 | no |
| -2 | 24 | no |
| 3 | 504 | no |
| -3 | 6 | no |
| 4 | 720 | no |
| -4 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -15 | -15 |
| Product of the roots | -120 | -120 |
Questions
What are the roots of x³ + 15x² + 74x + 120 = 0?
The solutions of x³ + 15x² + 74x + 120 = 0 are x = -6, -5, -4.
How do you factor x³ + 15x² + 74x + 120?
(x + 6)(x + 5)(x + 4).
What is the sum of the roots?
-15 (it equals −15 by Vieta's formulas).