Prove that 0 is a root of the equation $a_nx^n+\cdots a_1x+a_0=0$ if and only if the free term $a_0=0$.
Is this as simple as if $0$ is a root then $x=0$ is a solution and substituting it into the above equation you get $0+0+\cdots a_0=0$ so $a_0=0$. Also if $a_0=0$ then $a_nx+\cdots a_1x=0$ Then can factor out an $x$ and get $x=0$ so $0$ is a root of the polynomial.