Is $F$ a homomorphism?

Question

Let $F$ be the function $\mathbb R[x]$ to $\mathbb R$ by the rule $F(p(x)) = p(0)$.

Is $F$ a ring homomorphism from $\mathbb R[x]$ to $\mathbb R$?

Answer 1

The question boils down to how multiplication and addition of polynomials are defined. And those are defined pointwise: substituting any real number $r$ in place of the variable $x$ for polynomials $p$ and $q$, we have $$(p+q)(r)\ =\ p(r)+q(r)\\ (p\cdot q)(r)\ =\ p(r)\cdot q(r)\,.$$ In particular these hold for $r=0$.