Section+2.3

Section 2.3 Polynomial Functions of Higher Degree Recall: Polynomials fit the pattern: f(x)=AnX^n+An-1X^n-1+...+A1X+A0 n must be a positive integer Leading term: AnX^n Leading Coefficient: An Degree: n Degree-Name 3 -Cubic 4 -Quartic Combining monomial functions creates new graphs. Ex: f(x)=x^3-x Find any extrema: .384, -.384 Find any zeroes: x=1, x=-1, x=0 Properties of Polynomials: 1. Continuous 2. Smooth. (No corners or cusps) 3. If n is the degree of f, then f has at most n-1 local extrema and at most n zeroes.

End behavoir: What happens as x goes to positive of negative infinity? -Determined by the leading term.

Degree/Coefficient Even Odd Positive lim. x-->infinity=infinity lim. x--> infinity=infinity lim. x-->negative infinity=infinity lim. x-->negative infinity=negative infinity Negative lim x-->infintiy=neg. infinity limx--> infinity=neg infinity limx-->neg infinity=neg infinity limx-->neg infinity=infinity

Zeroes of polynomials functions: (x-intercept) Example: f(x)=x^3-x^2-6x 0=x^3-x^2-6x 0=x(x^2-x-6) 0=x(x-3)(x+2) x=3 x=-2

Multiplicity: If (x-c)^m is a factor of f then c is a zero with multiplicity m Example: f(x)=(x-2)^3(x+1)^2 0=(x-2)^3(x+1)^2 x=2 x=-1 mult. 3 mult. 2

Behavior at the x-int. If multiplicity is odd then it crosses the x-axis, if it is even, then it touches the x-axis Example: Graph: f(x)= (x+2)^3(x-1)^2 0=(x+2)^3(x-1)^2 x=-2 x=1 mult.3 mult. 2 cross touch

Homework: Page 259 problems #1-53, E.O.O.