Understanding Polynomial Functions in Mathematics
1. Definition
A polynomial function is a mathematical expression of the form:
f(x) = anxn + an-1xn-1 + ... + a1x + a0
where:
- an, an-1, ..., a0 are real numbers known as coefficients.
- n is a non-negative integer called the degree of the polynomial.
- an ≠ 0.
Examples of polynomial functions include:
- f(x) = 3x2 - 2x + 1 (degree 2)
- g(x) = -5x4 + 2x2 - 7 (degree 4)
- h(x) = 6 (constant polynomial, degree 0)
2. Degree of a Polynomial
The degree of a polynomial is the highest power of x with a non-zero coefficient. Here are some examples:
- Degree 0: constant function f(x) = 5
- Degree 1: linear function f(x) = 2x - 3
- Degree 2: quadratic function f(x) = 2x2 - 4x + 1
- Degree 3: cubic function f(x) = x3 + 2x
3. Domain and Range
The domain of any polynomial function is all real numbers: ℝ.
The range depends on the degree and the leading coefficient of the polynomial.
For example:
- f(x) = x2 has a range of [0, +∞)
- f(x) = x3 has a range of ℝ