Solve x⁴ - 4x³ - 26x² + 60x + 225 = 0
The solutions of x⁴ - 4x³ - 26x² + 60x + 225 = 0 are x = -3, -3, 5, 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 225, so the candidates are ±1, ±3, ±5, ±9, ±15, ±25, ±45, ±75, ±225.
- 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 - 10x + 25 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} - 4x^{3} - 26x^{2} + 60x + 225 = (x + 3)^{2}(x - 5)^{2} = 0 \]
\[ x^{4} - 4x^{3} - 26x^{2} + 60x + 225 = 0 \]
\[ \begin{array}{c|ccccc} -3 & 1 & -4 & -26 & 60 & 225 \\ & 0 & -3 & 21 & 15 & -225 \\ \hline & 1 & -7 & -5 & 75 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -3 & 1 & -7 & -5 & 75 \\ & 0 & -3 & 30 & -75 \\ \hline & 1 & -10 & 25 & 0 \end{array} \]
\[ x^{2} - 10x + 25 = 0 \Rightarrow x = 5 \]
\[ x^{4} - 4x^{3} - 26x^{2} + 60x + 225 = (x + 3)^{2}(x - 5)^{2} \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 256 | no |
| -1 | 144 | no |
| 3 | 144 | no |
| -3 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | 4 | 4 |
| Product of the roots | 225 | 225 |
Questions
What are the roots of x⁴ - 4x³ - 26x² + 60x + 225 = 0?
The solutions of x⁴ - 4x³ - 26x² + 60x + 225 = 0 are x = -3, -3, 5, 5 (repeated roots counted once each: -3, 5).
How do you factor x⁴ - 4x³ - 26x² + 60x + 225?
(x + 3)(x − 5).
What is the sum of the roots?
4 (it equals −(-4) by Vieta's formulas).