Solve x⁴ - 14x³ + 72x² - 162x + 135 = 0
The solutions of x⁴ - 14x³ + 72x² - 162x + 135 = 0 are x = 3, 3, 3, 5 (repeated roots counted once each: 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 135, so the candidates are ±1, ±3, ±5, ±9, ±15, ±27, ±45, ±135.
- 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 = 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 - 8x + 15 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} - 14x^{3} + 72x^{2} - 162x + 135 = (x - 3)^{3}(x - 5) = 0 \]
\[ x^{4} - 14x^{3} + 72x^{2} - 162x + 135 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & -14 & 72 & -162 & 135 \\ & 0 & 3 & -33 & 117 & -135 \\ \hline & 1 & -11 & 39 & -45 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 3 & 1 & -11 & 39 & -45 \\ & 0 & 3 & -24 & 45 \\ \hline & 1 & -8 & 15 & 0 \end{array} \]
\[ x^{2} - 8x + 15 = 0 \Rightarrow x = 5,\ 3 \]
\[ x^{4} - 14x^{3} + 72x^{2} - 162x + 135 = (x - 3)^{3}(x - 5) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 32 | no |
| -1 | 384 | no |
| 3 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | 14 | 14 |
| Product of the roots | 135 | 135 |
Questions
What are the roots of x⁴ - 14x³ + 72x² - 162x + 135 = 0?
The solutions of x⁴ - 14x³ + 72x² - 162x + 135 = 0 are x = 3, 3, 3, 5 (repeated roots counted once each: 3, 5).
How do you factor x⁴ - 14x³ + 72x² - 162x + 135?
(x − 3)(x − 5).
What is the sum of the roots?
14 (it equals −(-14) by Vieta's formulas).