Every degree $d$ polynomial $P$ in $\mathbb C[x_1,\cdots,x_n]$ has a factorization into a product of irreducible polynomials $P=S_1\cdots S_l$ (unique up to scalar multiplication). I have a confusion about the factors $S_1,\cdots,S_l$.
Take for example the polynomial $$P=x^5 + x^4 y - 2 x^3 y^2 - 2 x^2 y^3 + x y^4 + y^5 $$ It factors as $P=(x-y)^2(x+y)^3$.
If I write $P=P_1P_2$ where $P_1=(x-y)^2$ and $P_2=(x+y)^3$ then we see that $P_1=(x-y)^2=(x-y)(x-y)$ so $P_1$ is reducible and similarly for $P_2$.
Or may be we should write $P=Q_1Q_2Q_3Q_4Q_5$ where $Q_1=Q_2=x-y$ and $Q_3=Q_4=Q_5=x+y$ and now the factors $Q_i$ are all irreducible.
To summarize, in our example the factors $S_1,S_2,..., S_l$ alluded to in the factorization result are $Q_1,...,Q_5$ and not $P_1, P_2$. Is my understanding of the statement correct? Thanks for your helpp!