Polynomial Functions Defn Polynomial function The coefficients are

Polynomial Functions Defn: Polynomial function 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? yes no no yes

Polynomial Functions Defn: Degree of a Function The largest exponent in 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 3 8 5 12

Polynomial Functions Defn: Power function of Degree 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, a), & (1, a). 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, -a), & (1, a). 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 squiggle if the number is greater than 1.

Polynomial Functions Identify the zeros and their multiplicity 3 is a zero with a multiplicity of 1. Graph crosses the x-axis. -2 is a zero with a multiplicity of 3. Graph squiggles. -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. -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.

Polynomial Functions State and graph a possible function. -1 2 4