Math Help - [SOLVED] Proving that a polynomial is 0

1. [SOLVED] Proving that a polynomial is 0

Hi

Challenging (more or less) question. I put it in this subforum as to avoid giving hints on the way to solve it.
My friend told me a method, but I'm curious to know if there are others.

So we have $P$ a polynomial such that its degree $\text{deg}(P)\leq n$, $n \in \mathbb{N}^*$
If $\int_0^1 P(t)t^k ~dt=0 \quad \forall k \in \mathbb{N} ~:~ 0\leq k\leq n$,
prove that $P(x)=0$.

2. Salut

Let $P(t)=\sum_{k=0}^{n}\:a_k\:t^k$

$\int_0^1 P(t)t^k ~dt=0 \quad \forall k \in \mathbb{N} ~:~ 0\leq k\leq n$

$\sum_{k=0}^{n}\:a_k \int_0^1 P(t)t^k ~dt=0$

$\int_0^1 P(t)\:\sum_{k=0}^{n}\:a_k t^k ~dt=0$

$\int_0^1 P^2(t) ~dt=0$

But of course $P^2(t) \geq 0$

3. Very neat solution

This is how a teacher told my friend :
Define the scalar product $\langle x,y\rangle=\int_0^1 x(t)y(t) ~dt$
$y(t)=t^k$ belongs to the basis of $P[X]$
So we have $\langle x,y\rangle=0$, for any y in the basis.
Hence x=0.

I don't know which theorem was used though

4. Originally Posted by Moo
Very neat solution

This is how a teacher told my friend :
Define the scalar product $\langle x,y\rangle=\int_0^1 x(t)y(t) ~dt$
$y(t)=t^k$ belongs to the basis of $P[X]$
So we have $\langle x,y\rangle=0$, for any y in the basis.
Hence x=0.

I don't know which theorem was used though
Because it implies it is orthogonal to itself. (exactly what running-gag showed)

Remember that $
\left\langle {p_1,p_1} \right\rangle = 0 \Leftrightarrow p_1 = \bold 0
$
(the 0 polynomial ) by the definition of inner-product

Okay, let $
\mathcal{P}_n \left[ t \right]
$
be the space of polynomials of degree less or equal than $n$ over R. (our field will be R)

We define: $
\left\langle , \right\rangle :\mathcal{P}_n \left[ t \right] \times \mathcal{P}_n \left[ t \right] \to \mathbb{R}
$
by: $
\left\langle {a\left( t \right),b\left( t \right)} \right\rangle = \int_0^1 {a\left( t \right) \cdot b\left( t \right)dt}
$

You can check that this is an inner-product.

Now $
B = \left\{ {1,t,...,t^n } \right\}
$
is a basis of $
\mathcal{P}_n \left[ t \right]
$

And $
p\left( t \right)
$
is orthogonal to each of them, hence it's orthogonal to all vectors of $
\mathcal{P}_n \left[ t \right]
$
just see that $
\left\langle {p\left( t \right),b_0 + \sum\limits_{k = 1}^n {b_k \cdot t^k } } \right\rangle = b_0 \cdot \left\langle {p\left( t \right),1} \right\rangle + b_k \cdot \sum\limits_{k = 1}^n {\left\langle {p\left( t \right),t^k } \right\rangle }
$
. (in particular to itself and hence $p(t)\equiv 0$)

5. This problem has a nice generalization.
If $f:[0,1]\to \mathbb{R}$ is continous with $\int_0^1 f(x)x^t dx = 0$ for all $t=0,1,2,3,...$ then $f$ is identically zero.

6. Originally Posted by ThePerfectHacker
This problem has a nice generalization.
If $f:[0,1]\to \mathbb{R}$ is continous with $\int_0^1 f(x)x^t dx = 0$ for all $t=0,1,2,3,...$ then $f$ is identically zero.
it's a nice and simple application of the Weierstrass approximation theorem.