Prove: $\int_{0}^{\infty}f(x)g(x)e^{-x}dx$ Is An Inner Product Over $P_2[x]$

Question

Let $P_2[x]$ be the vector space of all real polynomial from degree $2$ or less, for all $f,g\in P_2[x]$

Prove: $\langle f,g\rangle=\int_{0}^{\infty}f(x)g(x)e^{-x}dx$ Is An Inner Product Over $P_2[x]$

So the properties of an inner product are:

1. $\langle v,v \rangle\geq 0$ and $\langle v,v \rangle= 0 \iff v\equiv 0$

2. $\langle v,u \rangle=\langle v,u \rangle$ In real inner space

3. $\langle v+u,h \rangle=\langle v,h \rangle+\langle u,h \rangle$

So for start I took a general 2 degree polynomial $ax^2+bx+c$

$\langle ax^2+bx+c,ax^2+bx+c \rangle=\int_{0}^{\infty}(ax^2+bx+c)^2e^{-x}dx$

Can it be solve via integration by parts, Or I could use a property of the inner product to solve it?

Answer 1

You have a nonnegative function under integral. What can you say about this function if the integral is equal to zero? If a polynomial has more roots than its degree, what can you say about the coefficients of this polynomial?