Question

\[\ \text{A monic polynomial f(x) with integer coefficients has the property:}\]
\[\ \large f(x) * f''(x) - (f')^2 = -6x^{10} + 600x \]
\[\ \text {Find f(2)}.\]

Answer 1

\[
f(x) = \sum_{k=0}^na_kx^k
\]

Answer 2

Consider the largest that \(n\) could be.

Answer 3

\(f\) is degree \(n\)
\(f'\) is degree \(n-1\)
\(f''\) is degree \(n-2\)

Answer 4

So \(ff''\) is degree \(n+n-2 = 2n-2\)
While \((f')^2\) is degree \(2(n-1) = 2n-2\)

Answer 5

it might be easier to just use differential equations.

Answer 6

This is actually making sense to me.

Answer 7

hm, how would that work?

Answer 8

Well \[
yy''-y'y' = -6x^{10}+600x
\]

Answer 9

I'm not completely sure how to sole this diff eq other than guessing and checking.

Answer 10

Anyway \(2n-2\leq 10\implies n\leq 6\)

Answer 11

Supposing:\[
f(x) = \sum_{k=0}^{n}a_kx^k
\]Then \[
f'(x) = \sum_{k=1}^{n}ka_kx^{k-1}
\]And \[
f''(x) = \sum_{k=2}^{n}(k)(k-1)a_kx^{k-2}
\]

Answer 12

We know that \((f')^2\) is even, so it can't contribute to the \(600x^1\) term

Answer 13

By even what I really should say is that all powers will be even.

Answer 14

Hmmm actually, maybe the best place to start is all the odd powers...

Answer 15

I may not be correct on the even powers thing.

Answer 16

I found something interesting though: \[
y^{(0)}y^{(2)} - y^{(1)}y^{(1)}=−6x^{10}+600x
\]Differentiate: \[
y^{(1)}y^{(2)}+y^{(0)}y^{(3)} - y^{(2)}y^{(1)}-y^{(1)}y^{(2)} = y^{(0)}y^{(3)} - y^{(2)}y^{(1)}
\]So \[
y^{(0)}y^{(3)} - y^{(2)}y^{(1)} = -60x^{9}+600
\]Differentiate again:\[
y^{(1)}y^{(3)} +y^{(0)}y^{(4)} - y^{(3)}y^{(1)}- y^{(2)}y^{(2)} = y^{(0)}y^{(4)} - y^{(2)}y^{(2)}
\]So \[
y^{(0)}y^{(4)} - y^{(2)}y^{(2)} = -540x^{8}
\]

Answer 17

From the first differentiation we get \[
a_0\times 3!a_3-2!a_2\times a_1=600
\]

Answer 18

Before that, we would have gotten \[
a_0\times 2!a_2-a_1\times a_1=0
\]

Answer 19

So \[
2a_0a_2=(a_1)^2
\]

Answer 20

\[
6a_0a_3-2a_2a_1=600
\]

Answer 21

Sham try 154

Answer 22

wio did you get 154 as a final answer? Jw

Answer 23

no. how did you get it?