Solve x⁴ - 8x³ + 24x² - 32x + 16 = 0
The solutions of x⁴ - 8x³ + 24x² - 32x + 16 = 0 are x = 2, 2, 2, 2 (repeated roots counted once each: 2).
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 16, so the candidates are ±1, ±2, ±4, ±8, ±16.
- 2
Test the candidates until one gives 0. x = 2 is a root. Divide by (x − 2) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 3
x = 2 is a root. Divide by (x − 2) with synthetic division; the bottom row is the quotient and a 0 remainder.
- 4
What is left is the quadratic x² - 4x + 4 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} - 8x^{3} + 24x^{2} - 32x + 16 = (x - 2)^{4} = 0 \]
\[ x^{4} - 8x^{3} + 24x^{2} - 32x + 16 = 0 \]
\[ \begin{array}{c|ccccc} 2 & 1 & -8 & 24 & -32 & 16 \\ & 0 & 2 & -12 & 24 & -16 \\ \hline & 1 & -6 & 12 & -8 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 2 & 1 & -6 & 12 & -8 \\ & 0 & 2 & -8 & 8 \\ \hline & 1 & -4 & 4 & 0 \end{array} \]
\[ x^{2} - 4x + 4 = 0 \Rightarrow x = 2 \]
\[ x^{4} - 8x^{3} + 24x^{2} - 32x + 16 = (x - 2)^{4} \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 1 | no |
| -1 | 81 | no |
| 2 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | 8 | 8 |
| Product of the roots | 16 | 16 |
Questions
What are the roots of x⁴ - 8x³ + 24x² - 32x + 16 = 0?
The solutions of x⁴ - 8x³ + 24x² - 32x + 16 = 0 are x = 2, 2, 2, 2 (repeated roots counted once each: 2).
How do you factor x⁴ - 8x³ + 24x² - 32x + 16?
(x − 2).
What is the sum of the roots?
8 (it equals −(-8) by Vieta's formulas).