Solution
= Solution
A point $p$ belongs to $V(I_1+I_2)$ exactly when every polynomial $f_1+f_2$, with $f_j\in I_j$, vanishes at $p$. This implies that every member of each ideal vanishes by taking the other summand to be zero, and the converse follows by addition. Hence
$$
\boxed{V(I_1+I_2)=V(I_1)\cap V(I_2)}.
$$
This is the first <vanishing loci of ideal sums and intersections> identity.