Solve x³ + 7x² - 24x - 180 = 0
The solutions of x³ + 7x² - 24x - 180 = 0 are x = -6, -6, 5 (repeated roots counted once each: -6, 5).
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 -180, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±9, ±10, ±12, ±15, ±18, ±20, ±30, ±36, ±45, ±60, ±90, ±180.
- 2
Test the candidates until one gives 0. x = 5 is a root. Divide by (x − 5) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
What is left is the quadratic x² + 12x + 36 = 0. Solve it (factoring or the quadratic formula):
- 4
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{3} + 7x^{2} - 24x - 180 = (x + 6)^{2}(x - 5) = 0 \]
\[ x^{3} + 7x^{2} - 24x - 180 = 0 \]
\[ \begin{array}{c|cccc} 5 & 1 & 7 & -24 & -180 \\ & 0 & 5 & 60 & 180 \\ \hline & 1 & 12 & 36 & 0 \end{array} \]
\[ x^{2} + 12x + 36 = 0 \Rightarrow x = -6 \]
\[ x^{3} + 7x^{2} - 24x - 180 = (x + 6)^{2}(x - 5) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | -196 | no |
| -1 | -150 | no |
| 2 | -192 | no |
| -2 | -112 | no |
| 3 | -162 | no |
| -3 | -72 | no |
| 4 | -100 | no |
| -4 | -36 | no |
| 5 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -7 | -7 |
| Product of the roots | 180 | 180 |
Questions
What are the roots of x³ + 7x² - 24x - 180 = 0?
The solutions of x³ + 7x² - 24x - 180 = 0 are x = -6, -6, 5 (repeated roots counted once each: -6, 5).
How do you factor x³ + 7x² - 24x - 180?
(x + 6)(x − 5).
What is the sum of the roots?
-7 (it equals −7 by Vieta's formulas).