College Algebra, 2013a

Chapter 3
Polynomial Functions
3.1 Graphs of Polynomials
Three of the families of functions studied thus far – constant, linear and quadratic – belong to
a much larger group of functions called polynomials. We begin our formal study of general
polynomials with a definition and some examples.
Definition 3.1. A polynomial function is a function of the form
f(x) =a n x n + a n−1 x n−1 + ...+ a 2 x 2 + a 1 x + a 0 ,
where a 0 , a 1 , ..., a n are real numbers and n ≥ 1 is a natural number.
polynomial function is (−∞, ∞).
The domain of a
There are several things about Definition 3.1 that may be off-putting or downright frightening. The
best thing to do is look at an example. Consider f(x) =4x 5 − 3x 2 +2x − 5. Is this a polynomial
function? We can re-write the formula for f as f(x) =4x 5 +0x 4 +0x 3 +(−3)x 2 +2x +(−5).
Comparing this with Definition 3.1, weidentifyn =5,a 5 =4,a 4 =0,a 3 =0,a 2 = −3, a 1 =2
and a 0 = −5. In other words, a 5 is the coefficient of x 5 , a 4 is the coefficient of x 4 , and so forth; the
subscript on the a’s merely indicates to which power of x the coefficient belongs. The business of
restricting n to be a natural number lets us focus on well-behaved algebraic animals. 1
Example 3.1.1. Determine if the following functions are polynomials. Explain your reasoning.
1. g(x) = 4+x3
x
2. p(x) =
4x + x3
x
3. q(x) =
4x + x3
x 2 +4
4. f(x) = 3√ x 5. h(x) =|x| 6. z(x) =0
1 Enjoy this while it lasts. Before we’re through with the book, you’ll have been exposed to the most terrible of
algebraic beasts. We will tame them all, in time.