Hints for Problem Set 6
Polynomials

Problem 3: As pointed out by Kumiko, there is a typo. It should be b₃x³.
A hint for this problem: The left hand side is the derivative of some function (the coefficients suggest that). So, the left hand side is p'(x), for some polynomial p(x). A couple of questions, then:

What is p(1) - p(-1)?
Geometrically, what does p'(x) = 0 mean about the graph of a function?
How does the Mean Value Theorem link the two above questions?

Problem 5: Set
p(x) = a x² + b x + c.
Then p''(x) = 2a. If a > 0, then the graph of p(x) is concave up; if a < 0, then the graph of p(x) is concave down. (If a = 0, the graph is neither concave up nor concave down; the graph is actually the graph of a straight line.) Let's introduce some new notation:

p(1) = A
p(p(1)) = p(A) = B
p(p(p(1))) = p(B) = C

We know these three points are on the parabola:

(1,A)
(A,B)
(B,C)

And we know:
1 < A < B < C
and so we can order the three points listed above from left to right:
(1,A) is to the left of (A,B) which is to the left of (B,C)
What information can we get about the second derivative from these three points? Well, if p''(x) > 0, then p'(x) is increasing, and vice versa. Is the slope of the secant line connecting (1,A) to (A,B) less than the slope of the secant line connecting (A,B) to (B,C)? And, if the answer to this question is yes, how does that force the second derivative to be positive?

So that is how I went about it. I haven't answered the last question yet; I think the fact that the coefficients a, b, c are integers comes into play, but I don't know how. There are probably a number of ways to deal with this.

Problem 6: How many maximums and minimums does this graph have? How are the number of maximums and minimums related to the number of roots?