Solve x³ + 3x² - 25x - 75 = 0
The solutions of x³ + 3x² - 25x - 75 = 0 are x = -5, -3, 5.
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 -75, so the candidates are ±1, ±3, ±5, ±15, ±25, ±75.
- 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
What is left is the quadratic x² - 25 = 0. Solve it (factoring or the quadratic formula):
- 4
So the polynomial factors as
LaTeX source of the formulas on this page
\[ x^{3} + 3x^{2} - 25x - 75 = (x + 5)(x + 3)(x - 5) = 0 \]
\[ x^{3} + 3x^{2} - 25x - 75 = 0 \]
\[ \begin{array}{c|cccc} -3 & 1 & 3 & -25 & -75 \\ & 0 & -3 & 0 & 75 \\ \hline & 1 & 0 & -25 & 0 \end{array} \]
\[ x^{2} - 25 = 0 \Rightarrow x = 5,\ -5 \]
\[ x^{3} + 3x^{2} - 25x - 75 = (x + 5)(x + 3)(x - 5) \]Candidates tested (computed)
| Candidate x | p(x) | Root? |
|---|---|---|
| 1 | -96 | no |
| -1 | -48 | no |
| 3 | -96 | no |
| -3 | 0 | root |
Vieta's formulas check the answer (computed)
| Relation | From the roots | From the coefficients |
|---|---|---|
| Sum of the roots | -3 | -3 |
| Product of the roots | 75 | 75 |
Questions
What are the roots of x³ + 3x² - 25x - 75 = 0?
The solutions of x³ + 3x² - 25x - 75 = 0 are x = -5, -3, 5.
How do you factor x³ + 3x² - 25x - 75?
(x + 5)(x + 3)(x − 5).
What is the sum of the roots?
-3 (it equals −3 by Vieta's formulas).