Solve x⁴ + 11x³ + 45x² + 81x + 54 = 0
The solutions of x⁴ + 11x³ + 45x² + 81x + 54 = 0 are x = -3, -3, -3, -2 (repeated roots counted once each: -3, -2).
How to work it out
- 1
A quartic has up to 4 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 = -2 is a root. Divide by (x + 2) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
x = -3 is a root. Divide by (x + 3) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 4
What is left is the quadratic x² + 6x + 9 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} + 11x^{3} + 45x^{2} + 81x + 54 = (x + 3)^{3}(x + 2) = 0 \]
\[ x^{4} + 11x^{3} + 45x^{2} + 81x + 54 = 0 \]
\[ \begin{array}{c|ccccc} -2 & 1 & 11 & 45 & 81 & 54 \\ & 0 & -2 & -18 & -54 & -54 \\ \hline & 1 & 9 & 27 & 27 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -3 & 1 & 9 & 27 & 27 \\ & 0 & -3 & -18 & -27 \\ \hline & 1 & 6 & 9 & 0 \end{array} \]
\[ x^{2} + 6x + 9 = 0 \Rightarrow x = -3 \]
\[ x^{4} + 11x^{3} + 45x^{2} + 81x + 54 = (x + 3)^{3}(x + 2) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 192 | no |
| -1 | 8 | no |
| 2 | 500 | no |
| -2 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -11 | -11 |
| Product of the roots | 54 | 54 |
Questions
What are the roots of x⁴ + 11x³ + 45x² + 81x + 54 = 0?
The solutions of x⁴ + 11x³ + 45x² + 81x + 54 = 0 are x = -3, -3, -3, -2 (repeated roots counted once each: -3, -2).
How do you factor x⁴ + 11x³ + 45x² + 81x + 54?
(x + 3)(x + 2).
What is the sum of the roots?
-11 (it equals −11 by Vieta's formulas).