If $K$ is a field then $\frac{K[x_1,x_2,...,x_n]}{(x_1-\alpha_1,...,x_i-\alpha_i)}\cong K[x_{i+1},...,x_n]$

Question

If $K$ is a field and $\alpha_1, \alpha_2, ..., \alpha_p \in K$ then $$\frac{K[x_1,x_2,...,x_n]}{(x_1-\alpha_1,...,x_i-\alpha_i)}\cong K[x_{i+1},...,x_n] $$ My Attempt
I simply tried to use Hilbert's Nullstellensatz theorem to prove it but couldn't proceed.

Answer 1

Hint: Prove that the map $\phi: K[x_1,x_2,\dots,x_n]\to K[x_{i+1},\dots,x_n]$ given by $\phi(f)=(f(\alpha_1,\dots,\alpha_i,x_{i+1},\dots,x_n)$ is a surjective homomorphism with kernel $(x_1-\alpha_1,...,x_i-\alpha_i)$.