Definition of a Polynomial Function

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Here a few examples of polynomial functions:

f(x) = 4x3 + 8x2 + 2x + 3

g(x) = 2.4x5 + 3.2x2 + 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 non-negative 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) = 2.4x5 + 3.2x2 + 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. So, we write a simple x instead of x1.



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 is equal to 1, and we usually do not show multiplications by 1. So, the variable x does not appear in the last term.



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


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