Definition of a Polynomial Function
Polynomial Functions Function Institute Math Contents Index Home
Here a few examples of polynomial functions:
f(x) = 4x3 + 8x2 + 2x + 3
g(x) = 2.5x5 + 5.2x2 + 7
h(x) = 3x2
i(x) = 22.6
Polynomial functions are functions that have this form:
f(x) = anxn + an-1xn-1 + ... + a1x + a0
The value of n must be an nonnegative integer. That is, it must be whole number; it is equal to zero or a positive integer.
The coefficients, as they are called, are an, an-1, ..., a1, a0. These are real numbers.
The degree of the polynomial function is the highest value for n where an is not equal to 0.
So, the degree of g(x) = 2.5x5 + 5.2x2 + 7 is 5.
Notice that the second to the last term in this form actually has x raised to an exponent of 1, as in:
f(x) = anxn + an-1xn-1 + ... + a1x1 + a0
Of course, usually we do not show exponents of 1.
Notice that the last term in this form actually has x raised to an exponent of 0, as in:
f(x) = anxn + an-1xn-1 + ... + a1x + a0x0
Of course, x raised to a power of 0 makes it equal to 1, and we usually do not show multiplications by 1.
So, in its most formal presentation, one could show the form of a polynomial function as:
f(x) = anxn + an-1xn-1 + ... + a1x1 + a0x0
Here are some polynomial functions; notice that the coefficients can be positive or negative real numbers.
f(x) = 2.4x5 + 1.7x2 - 5.6x + 8.1
f(x) = 4x3 + 5.6x
f(x) = 3.7x3 - 9.2x2 + 0.1x - 5.2