1. Polynomial Functions Defn: Polynomial function
In the form of: 𝑓 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 + ⋯ 𝑎 1 𝑥+ 𝑎 0 .
The coefficients are real numbers.
The exponents are non-negative integers.
The domain of the function is the set of all real numbers.
Are the following functions polynomials?
𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3
𝑔 𝑥 =2 𝑥 2 −4𝑥+ 𝑥 −2
yes no
𝑘 𝑥 = 2 𝑥 𝑥 5 +3𝑥
ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥) no
yes

2. Polynomial Functions Defn: Degree of a Function
The largest degree of the function represents the degree of the function.
The zero function (all coefficients and the constant are zero) does not have a degree.
State the degree of the following polynomial functions
𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3
𝑔 𝑥 =2 𝑥 5 −4 𝑥 3 +𝑥−2 3 5
ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥)
𝑘 𝑥 = 4𝑥 3 +6 𝑥 11 − 𝑥 10 + 𝑥 12 8 12

3. Polynomial Functions Defn: Power function of Degree n
In the form of: 𝑓 𝑥 =𝑎 𝑥 𝑛 .
The coefficient is a real number.
The exponent is a non-negative integer.
Properties of a Power Function w/ n a Positive EVEN integer
Even function  graph is symmetric with the y-axis.
The domain is the set of all real numbers.
The range is the set of all non-negative real numbers.
The graph always contains the points (0,0), (-1,1), & (1,1).
The graph will flatten out for x values between -1 and 1.

4. Polynomial Functions
Properties of a Power Function w/ n a Positive ODD integer
Odd function  graph is symmetric with the origin.
The domain and range are the set of all real numbers.
The graph always contains the points (0,0), (-1,-1), & (1,1).
The graph will flatten out for x values between -1 and 1.

5. Polynomial Functions Defn: Real Zero of a function
If f(r) = 0 and r is a real number, then r is a real zero of the function.
Equivalent Statements for a Real Zero
r is a real zero of the function.
r is an x-intercept of the graph of the function.
(x – r) is a factor of the function.
r is a solution to the function f(x) = 0

6. Polynomial Functions Defn: Multiplicity
The number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m).
Zero Multiplicity of an Even Number
The graph of the function touches the x-axis but does not cross it.
Zero Multiplicity of an Odd Number
The graph of the function crosses the x-axis, but will flatten out if the number is greater than 1.

7. Polynomial Functions Identify the zeros and their multiplicity
𝑓 𝑥 = 𝑥−3 𝑥+2 3
3 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
-2 is a zero with a multiplicity of 3.
Graph crosses the x-axis.
𝑔 𝑥 =5 𝑥+4 𝑥−7 2
-4 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
7 is a zero with a multiplicity of 2.
Graph touches the x-axis.
𝑔 𝑥 = 𝑥+1 (𝑥−4) 𝑥−2 2
-1 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
4 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
2 is a zero with a multiplicity of 2.
Graph touches the x-axis.

8. State and graph a possible function.
Polynomial Functions
State and graph a possible function.
𝑔 𝑥 = 𝑥+1 (𝑥−4) 𝑥−2 2 -1 2 4