Polynomial Functions Polynomial function The coefficients are real

Polynomial Functions 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 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 3 8 5 12

Polynomial Functions End Behavior of a Function

Polynomial Functions End Behavior of a Function

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 Turning Points The point where a function changes directions from increasing to decreasing or from decreasing to increasing. If a function has a degree of n, then it has at most n – 1 turning points. If the graph of a polynomial function has t number of turning points, then the function has at least a degree of t + 1. What is the most number of turning points the following polynomial functions could have? 3 -1 2 8 -1 7 5 -1 12 -1 4 11

What is Multiplicity? Multiplicity is the number of times a solution appears in a polynomial or the exponent that is outside the binomial.

y = (x-1)(x+2)(x-3)3 (x-1) Has the multiplicity of 1 because the exponent is 1 and a solution of 1, because when x – 1 = 0 is solved, x = 1 (x + 2) multiplicity of 1 and a solution of -2 (x – 3)2 multiplicity of 2 and a solution of 3

y = x(2 x - 3)3 (3 x + 1)2 (x) multiplicity of 1 a solution of 0 (2 x - 3)3 multiplicity of 3 and a solution of 3/2 (3 x + 1)2 multiplicity of 2 and a solution of -1/3

y = (5 x - 4)2(3 x-6) (5 x - 4)2 multiplicity of 2 a solution of 4/5 (3 x-6) multiplicity of 1 and a solution of 2

Rules of Multiplicity • Multiplicity of 1 will go straight through the xaxis. • Multiplicity of 2 will bounce of the x-axis. • Multiplicity of 3 will curve through the x-axis.

Polynomial Functions 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.

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 crosses the x-axis. -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. Line with negative slope Line with positive slope Parabola opening down

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