Mathematics

Problem

Let

Q(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+k

with

a\ne0

. Respond to each item with complete justifications. (a) Show that

Q^{(6)}(x)=0

for every real number

x

. (b) Derive an explicit expression for

Q^{(5)}(x)

and explain why it is constant. Express the constant in terms of

a

. (c) Suppose

Q^{(3)}(0)=60

and

Q^{(4)}(0)=0

. Determine the values of

a

b

, and

c

in terms of

d

and

e

, or explain why they cannot be uniquely determined from the given information.