Solve x⁴ + 14x³ + 69x² + 140x + 100 = 0
The solutions of x⁴ + 14x³ + 69x² + 140x + 100 = 0 are x = -5, -5, -2, -2 (repeated roots counted once each: -5, -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 100, so the candidates are ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100.
- 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 + 10x + 25 = 0. Solve it (factoring or the quadratic formula):
- 5
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{4} + 14x^{3} + 69x^{2} + 140x + 100 = (x + 5)^{2}(x + 2)^{2} = 0 \]
\[ x^{4} + 14x^{3} + 69x^{2} + 140x + 100 = 0 \]
\[ \begin{array}{c|ccccc} -2 & 1 & 14 & 69 & 140 & 100 \\ & 0 & -2 & -24 & -90 & -100 \\ \hline & 1 & 12 & 45 & 50 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -2 & 1 & 12 & 45 & 50 \\ & 0 & -2 & -20 & -50 \\ \hline & 1 & 10 & 25 & 0 \end{array} \]
\[ x^{2} + 10x + 25 = 0 \Rightarrow x = -5 \]
\[ x^{4} + 14x^{3} + 69x^{2} + 140x + 100 = (x + 5)^{2}(x + 2)^{2} \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | 324 | no |
| -1 | 16 | no |
| 2 | 784 | no |
| -2 | 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 | 100 | 100 |
Questions
What are the roots of x⁴ + 14x³ + 69x² + 140x + 100 = 0?
The solutions of x⁴ + 14x³ + 69x² + 140x + 100 = 0 are x = -5, -5, -2, -2 (repeated roots counted once each: -5, -2).
How do you factor x⁴ + 14x³ + 69x² + 140x + 100?
(x + 5)(x + 2).
What is the sum of the roots?
-14 (it equals −14 by Vieta's formulas).