Roots of cubic equations
Roots of Cubic Equations
Cubic equations are polynomial equations of the third degree, typically written in the form:
$$ ax^3 + bx^2 + cx + d = 0 $$
where ( a ), ( b ), ( c ), and ( d ) are constants, and ( a \neq 0 ).
Characteristics of Cubic Equations
- A cubic equation always has at least one real root. This is a consequence of the Intermediate Value Theorem.
- It can have either one real root and two complex conjugate roots or three real roots.
- The sum of the roots is equal to (-\frac{b}{a}), according to Vieta's formulas.
Finding the Roots of a Cubic Equation
There are several methods to find the roots of a cubic equation:
Factorization: If the cubic can be factored into a product of a linear factor and a quadratic factor, the roots can be found by setting each factor equal to zero.
Graphical Method: Plotting the cubic function and finding the points where the graph intersects the x-axis.
Numerical Methods: Such as the Newton-Raphson method or synthetic division.
Cardano's Formula: A general solution for cubic equations that can be used to find all roots, real or complex.
Cardano's Formula
To apply Cardano's formula, the cubic equation must first be reduced to a depressed cubic by a change of variable ( x = t - \frac{b}{3a} ). The depressed cubic has the form:
$$ t^3 + pt + q = 0 $$
Cardano's formula states that the roots of the depressed cubic are given by:
$$ t = \sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} $$
Table of Differences and Important Points
| Feature | Cubic Equations | Quadratic Equations |
|---|---|---|
| Degree | 3 | 2 |
| General Form | ( ax^3 + bx^2 + cx + d = 0 ) | ( ax^2 + bx + c = 0 ) |
| Number of Roots | Up to 3 (real or complex) | Up to 2 (real or complex) |
| Discriminant | ( \Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2 ) | ( \Delta = b^2 - 4ac ) |
| Real Roots | At least one | Zero, one, or two |
| Sum of Roots | ( -\frac{b}{a} ) | ( -\frac{b}{a} ) |
| Product of Roots | ( \frac{d}{a} ) (if all roots are real) | ( \frac{c}{a} ) |
Examples
Example 1: Factorization
Consider the cubic equation:
$$ x^3 - 6x^2 + 11x - 6 = 0 $$
This can be factored as:
$$ (x - 1)(x - 2)(x - 3) = 0 $$
Setting each factor to zero gives the roots:
$$ x = 1, x = 2, x = 3 $$
Example 2: Cardano's Formula
Consider the cubic equation:
$$ x^3 - 3x + 2 = 0 $$
First, we identify ( p = -3 ) and ( q = 2 ). Using Cardano's formula:
$$ t = \sqrt[3]{-\frac{2}{2} + \sqrt{\left(\frac{2}{2}\right)^2 + \left(\frac{-3}{3}\right)^3}} + \sqrt[3]{-\frac{2}{2} - \sqrt{\left(\frac{2}{2}\right)^2 + \left(\frac{-3}{3}\right)^3}} $$
Simplifying, we get:
$$ t = \sqrt[3]{-1 + \sqrt{1 - 1}} + \sqrt[3]{-1 - \sqrt{1 - 1}} $$
$$ t = \sqrt[3]{-1} + \sqrt[3]{-1} $$
$$ t = -1 + (-1) $$
$$ t = -2 $$
Since we did not change variables, ( t ) is the root of the original equation:
$$ x = -2 $$
The other two roots can be found by synthetic division or by using the quadratic formula on the remaining quadratic equation after factoring out ( (x + 2) ).
Understanding the roots of cubic equations is crucial for solving higher-degree polynomial equations and is a fundamental concept in algebra and precalculus.