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All of the functions that were graphed in Try This 1 are called polynomial functions. A polynomial function is any function that can be written as the sum or difference of terms where the variables have only positive integer exponents.

The formal definition for a polynomial function is any function that can be written in the following form: f(x) = anxn + an − 1xn − 1 + an − 2xn − 2 + … + a2x2 + a1x + a0, where

• n is a whole number
• x is a variable
• the coefficients a0, a1, a2, …, an are real numbers1

The following are all examples of polynomial functions:

• y = 3x + 1
• 
• k(x) = (2x − 1)(x + 3)

Although the last function, k(x), doesn’t look like it fits the definition, the key is that it can be written in the form shown above. Expanding k(x) results in

 

 

 

The last line is in the form described above; thus, k(x) fits the definition of a polynomial function.

 

Note that k(x) was originally written in factored form. It has two factors: 2x − 1 and x + 3.