1. ## Integrals problem 1

Let p(x) be a polynomial so that int(a-->a+1){p(x)dx} = 0 for all a in R.
Prove p(x)=0 for all x in R.

2. Originally Posted by Also sprach Zarathustra
Let p(x) be a polynomial so that int(a-->a+1){p(x)dx} = 0 for all a in R.
Prove p(x)=0 for all x in R.

Suppose $p(x)\neq 0$ and let $a\in\mathbb{R}$ be the largest real root of $p(x)$ (if the pol. has no real roots then the contradiction is immediate since then the pol's values are all positive or all negative), and get a contradiction evaluating the integral from a to a+1.

Tonio

3. "...and get a contradiction evaluating the integral from a to a+1."

How I get this kind of contradiction(for the maximality of a as root of p(x))?

My try:

let deg(p(x))=n
so int(a-->a+1){p(x)dx} = q(a+1) - q(a) = 0 , when deg(q(x))=n+1
so, deg{q(x+1) - q(x)}=n
q(a+1) - q(a)= m(a+1)=0
Thanks!

4. Originally Posted by Also sprach Zarathustra
Let p(x) be a polynomial so that int(a-->a+1){p(x)dx} = 0 for all a in R.
Prove p(x)=0 for all x in R.
So, $I_n=\sum_{j=1}^{n}\int_{j}^{j+1}p(x)dx=\int_{1}^{n +1}p(x)=0$. Thus, $\lim_{n\to\infty}I_n=0$. Thus, if $p(x)=a_0+\cdots+a_nx^n$ then $\int p(x)=a_0x+\cdots+ a_nx^{n+1}$ and so $\lim_{n\to\infty}\left(a_0x+\cdots+a_nx^{n+1}\righ t)=0$. Do your stuff.

5. WHY the coefficients p(x) and int(p(x)) are identical?

6. Originally Posted by Also sprach Zarathustra
WHY the coefficients p(x) and int(p(x)) are identical?
Oops. Stupid typo. Won't affect anything.

7. Originally Posted by Also sprach Zarathustra
"...and get a contradiction evaluating the integral from a to a+1."

How I get this kind of contradiction(for the maximality of a as root of p(x))?

Because after the largest real root the polynomial will be either all positive or all negative, so its integral cannot be zero, of course.

Tonio

My try:

let deg(p(x))=n
so int(a-->a+1){p(x)dx} = q(a+1) - q(a) = 0 , when deg(q(x))=n+1
so, deg{q(x+1) - q(x)}=n
q(a+1) - q(a)= m(a+1)=0