The biggest thing that it stopping me from progressing in this question is the fact that I have two elements in the ideal, this topic it still very new to me.
I can show that $I=\langle x \rangle$ and $I=\langle y \rangle$ are prime ideals on their own by showing that $^{K[x,y]}/_I$ is an integral domain through the First Isomorphism Theorem for Rings. How do I proceed with this question now that there are two elements in the ideal?
If I were to attempt it, knowing that $K$ is a field, I would define a homomorphism $\phi:K[x,y]\rightarrow K,f(x,y)\rightarrow f(0,0)$. Clearly $\ker \phi = \langle x \rangle \cup\langle y \rangle=\langle x,y\rangle$. So we have that $^{K[x,y]}/_{\langle x , y \rangle} \cong K$ by the First Isomorphism Theorem for Rings, and since $K$ is a field, it is an integral domain, and thus $\langle x , y \rangle$ is a prime ideal.
Best Answer
You can also try to factor out by the ideal $(x,y)$. $$\frac{K[x,y]}{(x,y)} \cong \frac{\frac{K[x,y]}{(x)}}{\frac{(x,y)}{(x)}} \cong \frac{K[y]}{(y)} \cong K$$