Higher-Order Polynomial Functions

Question

Given the polynomial function p(x)=−17(x+1)2(x−1)(x−3)2,p(x)=-17(x+1)2(x-1)(x-3)2, determine the following:

a) Degree: 

b) Left hand end behavior: As x→x→? -∞ ∞  , y→y→ ? -∞ ∞ 

c) Right hand end behavior: As x→x→? -∞ ∞  , y→y→ ? -∞ ∞ 

d) The degree of the polynomial is Select an answer even odd 

e) The yy-intercept is    .

f) How many turning points does the polynomial have? 

g) State the zeros from lowest xx-value to highest, and their multiplicity:

 with multiplicity 

 with multiplicity 

 with multiplicity 

h) Select the graph by using the characteristics from above:

1. 
   

Answer 1

p(x) = -17(x+1)^2(x-1)(x-3)^2

is a 5th degree polynomial, an odd degree

4 turning points, one less than the degree

with zeros -1, 1 and 3. -1 and 3 have multiplicity 2, 1 has multiplicity 1

multiplicity is the exponent of each factor

zeros are the x values that make each factor = 0

the y intercept is -17(9)(1)(-1) = 153, the constant term when x=0

right end behavior: approaches -infinity

due to -17 times (x+1) <0 when x>0

left end: approaches infinity

due to -17(x+1)>0 when x<1