Solve x³ - 3x² - 9x + 27 = 0
The solutions of x³ - 3x² - 9x + 27 = 0 are x = -3, 3, 3 (repeated roots counted once each: -3, 3).
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 27, so the candidates are ±1, ±3, ±9, ±27.
- 2
Test the candidates until one gives 0. x = 3 is a root. Divide by (x − 3) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
What is left is the quadratic x² - 9 = 0. Solve it (factoring or the quadratic formula):
- 4
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{3} - 3x^{2} - 9x + 27 = (x + 3)(x - 3)^{2} = 0 \]
\[ x^{3} - 3x^{2} - 9x + 27 = 0 \]
\[ \begin{array}{c|cccc} 3 & 1 & -3 & -9 & 27 \\ & 0 & 3 & 0 & -27 \\ \hline & 1 & 0 & -9 & 0 \end{array} \]
\[ x^{2} - 9 = 0 \Rightarrow x = 3,\ -3 \]
\[ x^{3} - 3x^{2} - 9x + 27 = (x + 3)(x - 3)^{2} \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 16 | no |
| -1 | 32 | no |
| 3 | 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 | -27 | -27 |
Questions
What are the roots of x³ - 3x² - 9x + 27 = 0?
The solutions of x³ - 3x² - 9x + 27 = 0 are x = -3, 3, 3 (repeated roots counted once each: -3, 3).
How do you factor x³ - 3x² - 9x + 27?
(x + 3)(x − 3).
What is the sum of the roots?
3 (it equals −(-3) by Vieta's formulas).