Section+2.3

//__Polynomial Functions of Higher Degree__//

Recall: Polynomial fits the pattern.   f(x)= anx2 + an-1xn-1 +....a,x+a0 Leading term: anx2 Leading coefficient: an Degree: n

__Combing Monomials creates new functions.__ Ex. f(x)=x3-x Find the extrema and zeros. 0=x3-x = x(x+1)(x-1) x= 0, 1, -1

__Properties of all polynomial functions__ 1. Continuous 2. Smooth (no corners or cusps) 3. If n is the degree of f, then f has at most x-1 extrema and at most n zeros.

__End Behavior__ Determined by the leading term

lim x-» -infinity = infinity || lim x-» infinity = -infinity lim x-» -infinity = -infinity || lim x-» -infinity = -infinity || lim x-» infinity = -infinity lim x-» -infinity = infinity ||
 * Coefficent////Degree || Even || Odd ||
 * Positive || lim x-» infinity = infinity
 * Negative || lim x-» infinity = -infinity

__Zeros of polynomial functions: (x-int)__ Ex. Find the zeros 0=x3-x2-6x =x(x2-x-6) =x(x-3)(x+2) x=0 x=3 x=-2

__Multiplicity__ If (x-c)m is a factor of f, then c, is a zero with multiplicity m Ex. f(x)= (x-2)3 (x+1)2 0=(x-2)3 (x+1)2 X=2 X=-1 Multi-3 Multi-2

__Behavior of the x-int__ If the x-int has //odd// multiplicity then the graph //crosses// the x-axis. If the x-in has //even// multiplicity then the graph //touches// the x-axis.