Solve x³ - 13x² + 54x - 72 = 0
The solutions of x³ - 13x² + 54x - 72 = 0 are x = 3, 4, 6.
How to work it out
- 1
A cubic has up to 3 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 -72, so the candidates are ±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±18, ±24, ±36, ±72.
- 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
What is left is the quadratic x² - 10x + 24 = 0. Solve it (factoring or the quadratic formula):
- 4
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{3} - 13x^{2} + 54x - 72 = (x - 3)(x - 4)(x - 6) = 0 \]
\[ x^{3} - 13x^{2} + 54x - 72 = 0 \]
\[ \begin{array}{c|cccc} 3 & 1 & -13 & 54 & -72 \\ & 0 & 3 & -30 & 72 \\ \hline & 1 & -10 & 24 & 0 \end{array} \]
\[ x^{2} - 10x + 24 = 0 \Rightarrow x = 6,\ 4 \]
\[ x^{3} - 13x^{2} + 54x - 72 = (x - 3)(x - 4)(x - 6) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | -30 | no |
| -1 | -140 | no |
| 2 | -8 | no |
| -2 | -240 | no |
| 3 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | 13 | 13 |
| Product of the roots | 72 | 72 |
Questions
What are the roots of x³ - 13x² + 54x - 72 = 0?
The solutions of x³ - 13x² + 54x - 72 = 0 are x = 3, 4, 6.
How do you factor x³ - 13x² + 54x - 72?
(x − 3)(x − 4)(x − 6).
What is the sum of the roots?
13 (it equals −(-13) by Vieta's formulas).