Multivariate polynomials with infinite number of zeros

Question

I need to prove the following:

Let $P(x_1,\cdots,x_r)$ be a polynomial. Then $P(\alpha_1,\cdots,\alpha_r) = 0$ for all $(\alpha_1,\cdots,\alpha_r) \in \mathbb{R}^r_{+}$ such that $\sum\limits^r_{j=1} \alpha_j = 1$ if and only if $P(.)$ is the zero polynomial.

I think I have proven it also, using induction, with $A.\Gamma$'s help on this question.

My question is, is there any existing result (perhaps involving fundamental theorem for multivariate polynomials, e.g.) from which this follows? I tried looking for it, found Bezout's Theorem, but not sure if and how this can be derived using it. Any help is appreciated. Thank you.

Edit: The wording of the above statement is not correct, as pointed out by others. I have posted a new question with the correct wording as suggested in question comments, here.

Answer 1

This is not correct. Consider the polynomial $$ P(x_1, x_2, \dots, x_r) = (1 - \sum_j x_j)^2 \, . $$

Answer 2

The if part is obvious. For the only if, first notice that given any $(x_1, \dots, x_r) \in \mathbb{R}^r_{+}$, we can use the homogeneity of $P$ to establish the value of $P$ at this point: $$P(x_1, \dots, x_r)=\lambda^d P(\alpha_1,\cdots,\alpha_r)=0,$$ where $\lambda = \displaystyle \sum_{i=1}^r x_i>0$, $\alpha_i = \dfrac {x_i} {\lambda} \; \forall i$ and $d$ is the degree of $P$ (assuming $P$ is nonzero).

Now, knowing that $P=0$ at the open set $\mathbb{R}^r_{+}$, we can choose some point $(u_1, \dots, u_r) \in \mathbb{R}^r_{+}$ and take any partial derivative of $P$ at this point (including mixed of any order) to find coefficients of $P$. For example, if $P$ would have the term $k x_{i_1}^{a_1} \dots x_{i_m}^{a_m}$, then $$k = \frac 1 {a_1! \dots a_m!} \frac{\partial^{a_1 + \dots + a_m}P(x_1, \dots, x_r)}{\partial x_{i_1}^{a_1} \dots \partial x_{i_m}^{a_m}} (u_1, \dots, u_r) = 0$$ because any other term, if exist, would become zero after such differentiation (here we use the homogeneity again) and value of the derivative is completely defined by values of $P$ at a small open neighbourhood of $(u_1, \dots, u_r)$.