1. 4.3: Polynomial Functions

2. Types of Polynomial Functions
I. Polynomial Functions
Types of Polynomial Functions
Degree
Name
Constant 1
Linear 2
Quadratic 3
Cubic 4
Quartic 5
Quintic

3. Typical polynomial graphs Cubic graphs Quartic graphs

4. lim 𝑥→∞ 𝑓(𝑥) lim 𝑥→−∞ 𝑓(𝑥) −∞ ∞ ∞ −∞ ∞ ∞ −∞ −∞
End behavior of polynomial functions.
Example 3: List the end behavior of the following functions
lim 𝑥→∞ 𝑓(𝑥)
Function
lim 𝑥→−∞ 𝑓(𝑥) −∞ ∞ ∞ −∞ ∞ ∞ −∞ −∞

5. Zeros of Polynomial Functions
Example 4: Find the zeroes of the function below.

6. B. Repeated Zeros: when a factor is repeated, it’s zero is called a repeated zero.
C. Definition: If f is a polynomial function and 𝑥−𝑐 𝑚 is a factor of f, then c is a zero of multiplicity m of f.

7. Example 5: Find the multiplicity of each of the zeros for the function below, then sketch a graph of the function.

8. Zeros of odd and even multiplicity.
If a polynomial function f has a real zero c of odd multiplicity, then the graph of f crosses the x-axis at (c,0) and the value of f changes sign at x = c.
If a polynomial function f has a real zero c of even multiplicity, then the graph of f does not cross the x-axis at (c,0) and the value of f does not changes sign at x = c.
𝑓 𝑥 = 𝑥 𝑥−2 3
𝐸𝑣𝑒𝑛 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦
𝑂𝑑𝑑 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦

9. Example 6: sketching the graph of a factored polynomial.