Let A be a bounded linear operator defined on a complex Hilbert space and let | A | = (A A) 1 2 {|A|=(A^{∗}A)^{frac{1}{2}}}. Among other refinements of the well-known numerical radius inequality w 2(A) ≤ 1 2∥A A + AA ∗ ∥, we show that w 2(A) ≤ 1 4w 2(| A| + i |A ∗ |) + 1 8∥| A | 2 + | A ∗ | 2 ∥+ 1 4w(| A || A ∗ |) ≤ 1 2∥A A + AA ∗ ∥. w. Also, we develop inequalities involving the numerical radius and the spectral radius for the sum of the product operators, from which we derive the inequalities w p(A) ≤ 1 2w(| A | p + i| A ∗ | p) ≤ ∥A ∥p w^{p}(A)leqfrac{1}{sqrt{2}}w(|A|^{p}+{rm i}|A^{∗}|^{p})leq|A|^{p} for all p ≥ 1 {pgeq 1}. Further, we derive new bounds for the zeros of complex polynomials.