Solve x⁴ + 5x³ + x² - 21x - 18 = 0
The solutions of x⁴ + 5x³ + x² - 21x - 18 = 0 are x = -3, -3, -1, 2 (repeated roots counted once each: -3, -1, 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 -18, so the candidates are ±1, ±2, ±3, ±6, ±9, ±18.
- 2
Test the candidates until one gives 0. x = -1 is a root. Divide by (x + 1) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
x = 2 is a root. Divide by (x − 2) 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
\[ x^{4} + 5x^{3} + x^{2} - 21x - 18 = (x + 3)^{2}(x + 1)(x - 2) = 0 \]
\[ x^{4} + 5x^{3} + x^{2} - 21x - 18 = 0 \]
\[ \begin{array}{c|ccccc} -1 & 1 & 5 & 1 & -21 & -18 \\ & 0 & -1 & -4 & 3 & 18 \\ \hline & 1 & 4 & -3 & -18 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 2 & 1 & 4 & -3 & -18 \\ & 0 & 2 & 12 & 18 \\ \hline & 1 & 6 & 9 & 0 \end{array} \]
\[ x^{2} + 6x + 9 = 0 \Rightarrow x = -3 \]
\[ x^{4} + 5x^{3} + x^{2} - 21x - 18 = (x + 3)^{2}(x + 1)(x - 2) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | -32 | no |
| -1 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -5 | -5 |
| Product of the roots | -18 | -18 |
Questions
What are the roots of x⁴ + 5x³ + x² - 21x - 18 = 0?
The solutions of x⁴ + 5x³ + x² - 21x - 18 = 0 are x = -3, -3, -1, 2 (repeated roots counted once each: -3, -1, 2).
How do you factor x⁴ + 5x³ + x² - 21x - 18?
(x + 3)(x + 1)(x − 2).
What is the sum of the roots?
-5 (it equals −5 by Vieta's formulas).