Solve x⁴ + 9x³ - 5x² - 225x - 500 = 0

The solutions of x⁴ + 9x³ - 5x² - 225x - 500 = 0 are x = -5, -5, -4, 5 (repeated roots counted once each: -5, -4, 5).

x^{4} + 9x^{3} - 5x^{2} - 225x - 500 = (x + 5)^{2}(x + 4)(x - 5) = 0

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Graph of y = x⁴ + 9x³ - 5x² - 225x - 500. The curve crosses the x-axis at x = -5, x = -4, x = 5, the roots found above.

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 -500, so the candidates are ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100, ±125, ±250, ±500.
$x^{4} + 9x^{3} - 5x^{2} - 225x - 500 = 0$
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.
$\begin{array}{c|ccccc} -4 & 1 & 9 & -5 & -225 & -500 \\ & 0 & -4 & -20 & 100 & 500 \\ \hline & 1 & 5 & -25 & -125 & 0 \end{array}$
3
x = 5 is a root. Divide by (x − 5) with synthetic division; the bottom row is the quotient and a 0 remainder.
$\begin{array}{c|cccc} 5 & 1 & 5 & -25 & -125 \\ & 0 & 5 & 50 & 125 \\ \hline & 1 & 10 & 25 & 0 \end{array}$
4
What is left is the quadratic x² + 10x + 25 = 0. Solve it (factoring or the quadratic formula):
$x^{2} + 10x + 25 = 0 \Rightarrow x = -5$
5
So the polynomial factors as
$x^{4} + 9x^{3} - 5x^{2} - 225x - 500 = (x + 5)^{2}(x + 4)(x - 5)$

LaTeX source of the formulas on this page

\[ x^{4} + 9x^{3} - 5x^{2} - 225x - 500 = (x + 5)^{2}(x + 4)(x - 5) = 0 \]
\[ x^{4} + 9x^{3} - 5x^{2} - 225x - 500 = 0 \]
\[ \begin{array}{c|ccccc} -4 & 1 & 9 & -5 & -225 & -500 \\ & 0 & -4 & -20 & 100 & 500 \\ \hline & 1 & 5 & -25 & -125 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 5 & 1 & 5 & -25 & -125 \\ & 0 & 5 & 50 & 125 \\ \hline & 1 & 10 & 25 & 0 \end{array} \]
\[ x^{2} + 10x + 25 = 0 \Rightarrow x = -5 \]
\[ x^{4} + 9x^{3} - 5x^{2} - 225x - 500 = (x + 5)^{2}(x + 4)(x - 5) \]

Candidates tested (computed)

Candidate x	p(x)	Root?
1	-720	no
-1	-288	no
2	-882	no
-2	-126	no
4	-648	no
-4	0	root

Vieta's formulas check the answer (computed)

Relation	From the roots	From the coefficients
Sum of the roots	-9	-9
Product of the roots	-500	-500

Questions

What are the roots of x⁴ + 9x³ - 5x² - 225x - 500 = 0?

The solutions of x⁴ + 9x³ - 5x² - 225x - 500 = 0 are x = -5, -5, -4, 5 (repeated roots counted once each: -5, -4, 5).

How do you factor x⁴ + 9x³ - 5x² - 225x - 500?

(x + 5)(x + 4)(x − 5).

What is the sum of the roots?

-9 (it equals −9 by Vieta's formulas).

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