Solve x³ + 12x² + 45x + 54 = 0
The solutions of x³ + 12x² + 45x + 54 = 0 are x = -6, -3, -3 (repeated roots counted once each: -6, -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 54, so the candidates are ±1, ±2, ±3, ±6, ±9, ±18, ±27, ±54.
- 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 + 9x + 18 = 0. Solve it (factoring or the quadratic formula):
- 4
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{3} + 12x^{2} + 45x + 54 = (x + 6)(x + 3)^{2} = 0 \]
\[ x^{3} + 12x^{2} + 45x + 54 = 0 \]
\[ \begin{array}{c|cccc} -3 & 1 & 12 & 45 & 54 \\ & 0 & -3 & -27 & -54 \\ \hline & 1 & 9 & 18 & 0 \end{array} \]
\[ x^{2} + 9x + 18 = 0 \Rightarrow x = -3,\ -6 \]
\[ x^{3} + 12x^{2} + 45x + 54 = (x + 6)(x + 3)^{2} \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 112 | no |
| -1 | 20 | no |
| 2 | 200 | no |
| -2 | 4 | no |
| 3 | 324 | no |
| -3 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -12 | -12 |
| Product of the roots | -54 | -54 |
Questions
What are the roots of x³ + 12x² + 45x + 54 = 0?
The solutions of x³ + 12x² + 45x + 54 = 0 are x = -6, -3, -3 (repeated roots counted once each: -6, -3).
How do you factor x³ + 12x² + 45x + 54?
(x + 6)(x + 3).
What is the sum of the roots?
-12 (it equals −12 by Vieta's formulas).