Is there a way of separating symmetric multilinear forms?

Question

I have a very simple question which seemingly falls into the category of multilinear algebra. I need this concept in one of my research papers, but unfortunately did not learn it before. Let me state a simpler version of the question first:

Suppose that $\mathbf{A} := ((A_{i_1,\ldots,i_k}))_{1\leq i_1,\ldots,i_k\leq n}$ be a $k$-tensorof dimension $n\times \ldots\times n$. Assume that $A_{i_1,\ldots,i_k} = 0$ if $i_s = i_t$ for some $1\leq s<t\leq k$, and also assume that $\mathbf{A}$ is symmetric, i.e. $A_{i_1,\ldots,i_k} = A_{\pi(i_1)},\ldots,A_{\pi(i_k)}$ for any $1\leq i_1 <\ldots < i_k \leq n$ and any permutation $\pi$ on $\{1,\ldots,n\}.$ For any $n$-dimensional vector $\mathbf{x}:= (x_1,\ldots,x_n)$, define:

$$Q(\mathbf{x}) := \sum_{1\leq i_1,\ldots,i_k\leq n}A_{i_1,\ldots,i_k} x_{i_1}\ldots x_{i_k}~.$$ My question is: Is there a way to write the multilinear form $Q(\mathbf{x})$ as a weighted sum of powers of the entries of some transformed vector of $\mathbf{x}$? That is, can $Q(\mathbf{x})$ be written as: $$Q(\mathbf{x}) = \sum_{j=1}^n a_j y_j^m$$ for some $m$ (I am not sure whether this $m$ should be $k$ or $2$, or even something else, at this moment), where $\mathbf{y}$ is some transformed vector obtained from $\mathbf{x}$? In other words, I am looking for a spectral-type decomposition of the symmetric tensor $\mathbf{A}$, that allows one to write such $k$-multilinear forms in a separated manner.

Now, my original question is essentially the same as the one I asked just now, just in functional terminologies. I had a symmetric function $f: [0,1]^k \mapsto [0,1]$ and a function $g: [0,1]\mapsto [0,1]$, and I was trying to look at a way of expressing $$\int_{[0,1]^k} f(x_1,\ldots,x_k) g(x_1)\ldots g(x_k) dx_1\ldots dx_k$$ as a separated form similar to the one above, possibly in terms of the eigenobjects of the function $f$. I believe the first answer would lead to the second.

By the way, I tried to study some papers which treat spectral theory for tensors (such as https://arxiv.org/pdf/1008.2923.pdf and https://www.stat.uchicago.edu/~lekheng/work/davis.pdf), but could not quite come up with a solution. Any help will be appreciated, and thanks in advance!