JUN-SEOP SONG 4 two modified Newton’s methods combined with the derivative-free method. They are constructed by applying the derivative-free method to the x ∗ in the trapezoidal Newton’s method and the midpoint Newton’s method, respectively. In Section 4, we give a numer- ical comparison between various simultaneous methods for finding zeros of a polynomial. Finally, we conclude that the convergence of all new constructed methods in this paper are similar or superior than other iterative methods of cubic order. In this section, we construct a new method for finding simultaneously all simple zeros of polynomials of cubic order. By applying the Weierstrass method (5) to the x ∗ in the trapezoidal Newton’s method (3), we derive a new method constructed as follows: We call (7) the Weierstrass-like trapezoidal Newton’s method , and from this, simply, call it Method 1 . The calculation and discussion of the order of Method 1 are similar to those of the Newton-Weierstrass method, which is an alteration of Petkovi´ c’s midpoint Newton’s method (see [5]). From (7), we have the following theorem. Lemma 1. For a polynomial P ( z ) , we have Proof. By the Taylor’s expansion around r$_{i}$ , we have that II. Weierstrass-like Trapezoidal Newton’s method ˆ z$_{i}$ = z$_{i}$ − 2 P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) + P $^{′}$( z$_{i}$ − W$_{i}$ ) . (7) P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) ( P $^{′′}$( z$_{i}$ ) P ( z$_{i}$ ) 2 P $^{′}$( z$_{i}$ ) 2 O$_{M}$ ( e ) + O$_{M}$ ( e $^{2}$) ) = O$_{M}$ ( e $^{3}$) . P ( z$_{i}$ ) = P $^{′}$( r$_{i}$ ) ( e$_{i}$ + P $^{′′}$( r$_{i}$ ) 2 P $^{′}$( r$_{i}$ ) e 2 i + O$_{M}$ ( e $^{3}$) ) , P $^{′}$( z$_{i}$ ) = P $^{′}$( r$_{i}$ ) 1 + P $^{′′}$( r$_{i}$ ) P $^{′}$( r$_{i}$ ) e$_{i}$ + P $^{′′′}$( r$_{i}$ ) 2 P $^{′}$( r$_{i}$ ) e 2 i + O$_{M}$ ( e $^{3}$) ) , (8) P $^{′′}$( z$_{i}$ ) = P $^{′}$( r$_{i}$ ) ( P $^{′′}$( r$_{i}$ ) P $^{′}$( r$_{i}$ ) + P $^{′′′}$( r$_{i}$ ) P $^{′}$( r$_{i}$ ) e$_{i}$ + P $^{′′′′}$( r$_{i}$ ) 2 P $^{′}$( r$_{i}$ ) e 2 i + O$_{M}$ ( e $^{3}$) ) .