Solve x⁴ - 15x³ + 84x² - 208x + 192 = 0
The solutions of x⁴ - 15x³ + 84x² - 208x + 192 = 0 are x = 3, 4, 4, 4 (repeated roots counted once each: 3, 4).
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 192, so the candidates are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±32, ±48, ±64, ±96, ±192.
- 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 - 8x + 16 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} - 15x^{3} + 84x^{2} - 208x + 192 = (x - 3)(x - 4)^{3} = 0 \]
\[ x^{4} - 15x^{3} + 84x^{2} - 208x + 192 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & -15 & 84 & -208 & 192 \\ & 0 & 3 & -36 & 144 & -192 \\ \hline & 1 & -12 & 48 & -64 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 4 & 1 & -12 & 48 & -64 \\ & 0 & 4 & -32 & 64 \\ \hline & 1 & -8 & 16 & 0 \end{array} \]
\[ x^{2} - 8x + 16 = 0 \Rightarrow x = 4 \]
\[ x^{4} - 15x^{3} + 84x^{2} - 208x + 192 = (x - 3)(x - 4)^{3} \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 54 | no |
| -1 | 500 | no |
| 2 | 8 | no |
| -2 | 1080 | no |
| 3 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | 15 | 15 |
| Product of the roots | 192 | 192 |
Questions
What are the roots of x⁴ - 15x³ + 84x² - 208x + 192 = 0?
The solutions of x⁴ - 15x³ + 84x² - 208x + 192 = 0 are x = 3, 4, 4, 4 (repeated roots counted once each: 3, 4).
How do you factor x⁴ - 15x³ + 84x² - 208x + 192?
(x − 3)(x − 4).
What is the sum of the roots?
15 (it equals −(-15) by Vieta's formulas).