Exploring the Solutions of a Third Degree Equation

Before diving into the topic, let's first revisit the solutions of first and second degree equations, which serve as a foundation for understanding third degree equations.

Solutions of First and Second Degree Equations

First degree equations have exactly one solution:

7x - 2 58

x 8

Second degree equations, or quadratic equations, can have zero, one, or two real solutions based on the discriminant:

For x2 - 8 4, there are zero real solutions.

For x2 - 4x - 4 0, there is one real solution, which is x -2.

For x2 - 3x 2 0, there are two real solutions, x 1 and x 2.

Guessing the Number of Solutions for a Third Degree Equation

Given the examples listed, it's natural to wonder how many real solutions a third degree equation could have. While a first degree equation has one solution and a second degree equation can have zero, one, or two solutions, the question of solutions for a third degree equation might not be as straightforward. Let's explore this.

Possible Solutions for a Third Degree Equation

A third degree equation, or cubic equation, can indeed have one, two, or three real solutions. However, some solutions might be repeated. Let's delve into the examples and reasoning behind these possibilities.

First, consider a simple homogeneous polynomial:

y 2x3

This equation represents a cubic function that touches the x-axis at the origin (0,0) but does not cross it. The graph indicates an infinite number of solutions along the x-axis for this specific function. However, this is a special case and not the general form of a third degree equation.

Similarly, third degree equations that are differential equations with insufficient boundary or initial conditions might also have an infinite number of solutions. In these cases, the integration constants are not fully determined, leading to multiple possible solutions.

General Case: Polynomial Equations of the 3rd Degree

For a general third degree polynomial equation with real coefficients, the Fundamental Theorem of Algebra states that there are exactly three roots, which can be real or complex. However, all complex roots occur in conjugate pairs. Therefore, a polynomial equation of the third degree will have either one real root and a pair of complex conjugate roots, or it will have three real roots (one of which may be a repeated root).

Graphical Interpretation

To visualize these solutions, one can use a graphing calculator to plot the equation. For example, consider the following equations in factored form:

Equation with one real solution: x3 - 4x2 4x 0

The graph would cross the x-axis at one point, indicating a single real solution.

Equation with two real solutions: x3 - 6x2 11x - 6 0

The graph would cross the x-axis at two points, indicating two real solutions.

Equation with three real solutions: x3 2x2 - 8x - 12 0

The graph would cross the x-axis at three points, indicating three real solutions.

Conclusion

Based on the Fundamental Theorem of Algebra, a third degree polynomial equation will always have three roots. However, the nature of these roots (real or complex) and whether they are distinct or repeated depends on the specific coefficients of the equation. In most practical cases, we use real-world applications where we are interested in the real roots of the equation. Thus, for the purposes of solving real-world problems, we typically look for three real solutions.

Additional Resources

For further exploration, you may want to refer to the following resources:

WolframAlpha: Solve x3 - 4x2 4x 0

Khan Academy: Roots of polynomials

Coursera: Solving polynomial and cubic equations

Through these varied resources, you can deepen your understanding of third degree equations and their solutions.