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 3 +3 4 𝑥 5 +3𝑥 ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥) no yes

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

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.

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.

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

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.

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.

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