Solve x³ - 12x² + 48x - 64 = 0
The solutions of x³ - 12x² + 48x - 64 = 0 are x = 4, 4, 4 (repeated roots counted once each: 4).
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 -64, so the candidates are ±1, ±2, ±4, ±8, ±16, ±32, ±64.
- 2
Test the candidates until one gives 0. x = 4 is a root. Divide by (x − 4) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
What is left is the quadratic x² - 8x + 16 = 0. Solve it (factoring or the quadratic formula):
- 4
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{3} - 12x^{2} + 48x - 64 = (x - 4)^{3} = 0 \]
\[ x^{3} - 12x^{2} + 48x - 64 = 0 \]
\[ \begin{array}{c|cccc} 4 & 1 & -12 & 48 & -64 \\ & 0 & 4 & -32 & 64 \\ \hline & 1 & -8 & 16 & 0 \end{array} \]
\[ x^{2} - 8x + 16 = 0 \Rightarrow x = 4 \]
\[ x^{3} - 12x^{2} + 48x - 64 = (x - 4)^{3} \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | -27 | no |
| -1 | -125 | no |
| 2 | -8 | no |
| -2 | -216 | no |
| 4 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | 12 | 12 |
| Product of the roots | 64 | 64 |
Questions
What are the roots of x³ - 12x² + 48x - 64 = 0?
The solutions of x³ - 12x² + 48x - 64 = 0 are x = 4, 4, 4 (repeated roots counted once each: 4).
How do you factor x³ - 12x² + 48x - 64?
(x − 4).
What is the sum of the roots?
12 (it equals −(-12) by Vieta's formulas).