Commutative ring with no non-zero nilpotent elements

Question

Assignment:

Let R be a commutative ring with no non-zero nilpotent elements ( that is, an = 0  implies).
If f(x) = a₀ + a₁x + .... + a_mx^m in R[x] is a zero-divisor,
prove that there is an element b ≠ 0 in R such that ba₀ = ba₁ = ba₂ =.....=ba_m = 0

Provide complete and step by step solution for the question and show calculations and use formulas.