Solve x⁴ - 33x² - 8x + 240 = 0
The solutions of x⁴ - 33x² - 8x + 240 = 0 are x = -4, -4, 3, 5 (repeated roots counted once each: -4, 3, 5).
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 240, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±16, ±20, ±24, ±30, ±40, ±48, ±60, ±80, ±120, ±240.
- 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
x = -4 is a root. Divide by (x + 4) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 4
What is left is the quadratic x² - x - 20 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} - 33x^{2} - 8x + 240 = (x + 4)^{2}(x - 3)(x - 5) = 0 \]
\[ x^{4} - 33x^{2} - 8x + 240 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & 0 & -33 & -8 & 240 \\ & 0 & 3 & 9 & -72 & -240 \\ \hline & 1 & 3 & -24 & -80 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -4 & 1 & 3 & -24 & -80 \\ & 0 & -4 & 4 & 80 \\ \hline & 1 & -1 & -20 & 0 \end{array} \]
\[ x^{2} - x - 20 = 0 \Rightarrow x = 5,\ -4 \]
\[ x^{4} - 33x^{2} - 8x + 240 = (x + 4)^{2}(x - 3)(x - 5) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 200 | no |
| -1 | 216 | no |
| 2 | 108 | no |
| -2 | 140 | no |
| 3 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | 0 | 0 |
| Product of the roots | 240 | 240 |
Questions
What are the roots of x⁴ - 33x² - 8x + 240 = 0?
The solutions of x⁴ - 33x² - 8x + 240 = 0 are x = -4, -4, 3, 5 (repeated roots counted once each: -4, 3, 5).
How do you factor x⁴ - 33x² - 8x + 240?
(x + 4)(x − 3)(x − 5).
What is the sum of the roots?
0 (it equals −0 by Vieta's formulas).