Let p(z) = zn+ an-1zn-1+ an-2zn-2+ ⋯ + a1z+ a be a complex polynomial with a≠ 0 and n≥ 3. Several new upper bounds for the moduli of the zeros of p are developed. In particular, if α=∑j=0n-1|aj|2 and z is any zero of p, then we show that |z|2≤cos2πn+1+|an-2|+14(|an-1|+α)2+12α2-|an-1|2+12α,which is sharper than the existing bound, given as, |z|2≤cos2πn+1+14(|an-1|+α)2+α,if and only if 2|an-2|<∑j=0n-1|aj|2-∑j=0n-2|aj|2. The upper bounds obtained here enable us to describe smaller annuli in the complex plane containing all the zeros of p.