Prove that bilinear form can be presented as product of linear forms if and only if it has rank one

Let $f: V \rightarrow F$ a bilinear form such that $f \ne 0$. Prove that $f$ can be presented as a product of two linear forms $f(X,Y) = \left(\sum\limits_i^nb_ix_i \right)\ast \left(\sum\limits_j^nc_jy_j\right)$ if and only if $rank(f)=1$

I saw this question and it's answer: Prove that the bilinear form can be presented as a product of two linear forms

But the answer seems to ignore the fact that $rank(f)$ must be $1$. I don't know how to use that fact to solve this.

Thanks in advance

The source of your confusion might be the fact that the answer in the linked question does not actually answer the question. In fact, this answer shows that if a bilinear form is the product of two linear forms, then it has rank one (just observe that the matrix in the final equation has rank one). So it covers one implication of your question (but not the one that was asked in the linked question).

For the other implication, all you need to do is show that a rank one matrix must be a product of a column matrix and a row matrix. For this, simply observe that if $A$ has rank one, then all its column must be multiple of one another ; so you can choose a column that is non-zero, and all the others will be multiple of that column. Hence multiplying it by the row matrix whose elements are the coefficients of each column gives you the matrix $A$.