A third degree polynomial, also known as a cubic polynomial, is a mathematical expression of the form:
f(x) = ax^3 + bx^2 + cx + d
In this equation, a, b, c, and d are coefficients and a cannot be zero (if a is zero, then the polynomial will not be of degree three). The variables represent:
- x: the variable or input of the polynomial
- f(x): the output of the polynomial, which is the value of the polynomial at a given x
Key characteristics of third degree polynomials include:
- Degree: The highest power of the variable x is 3, which defines it as a third degree polynomial.
- Graph Shape: When graphed, the polynomial can have one of the following shapes: it can cross the x-axis up to three times (indicating three real roots), or less if some of the roots are complex or repeated. The graph has an ‘S’ shape, which allows for varying behavior.
- Turning Points: A third degree polynomial can have a maximum of two turning points, where the direction of the graph changes from increasing to decreasing or vice versa.
- End Behavior: In terms of the ends of the graph, as x approaches infinity, f(x) will approach either positive or negative infinity depending on the sign of a. If a is positive, the graph will rise to positive infinity on the right end and fall to negative infinity on the left end; if a is negative, it will do the opposite.
Examples of third degree polynomials include:
- f(x) = 2x^3 + 3x^2 – 5x + 1
- g(x) = -x^3 + x – 4
These polynomials can be analyzed further in terms of their roots (solutions), behavior, and applications in various fields, including physics, engineering, and economics. Understanding third degree polynomials is essential for higher mathematics and calculus, making them a fundamental topic in algebra.