NEW SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A POLYNOMIAL 3 Both the trapezoidal Newton’s method and the midpoint Newton’s method are of cubic order, while the original Newton’s method was of quadratic order. A variety of methods can be applied to the x ∗ in addition to Newton’s method. Petkovi´ c [5] derived the fol- lowing simultaneous method for finding all simple zeros of polynomials by applying the Weierstrass method to the x ∗ in the midpoint Newton’s method: which has a similar form with the one above and this method is of cubic order. In this paper, we present three new methods for the simultaneous approximation of all simple zeros of polynomials by applying the Weierstrass-like method and the derivative- free method to x ∗ in the trapezoidal Newton’s method and the midpoint Newton’s method. which is called Newton-Weierstrss method (or NWM). Also, Petkovi´ c [6] found the fol- lowing derivative-free method (or DFM) defined as ˆ z$_{i}$ = z$_{i}$ − P ( z$_{i}$ ) P ′ ( z$_{i}$ − 1 $_{2}$W$_{i}$ ) , (5) ˆ z$_{i}$ = z$_{i}$ − W$_{i}$ 1 − P ( z$_{i}$ − W$_{i}$ ) /P ( z$_{i}$ ) , (6) Throughout this paper, the convergence of zeros will be discussed and the order will be calculated for new constructed methods. We will use the notation a = O$_{M}$ ( b ) for two complex numbers a and b whose moduli are of the same order, that is, | a | = O ( | b | ). In addition, the error is defined as | e | = max i =1 $_{,...,n}${ | e$_{i}$ |} with e$_{i}$ = z$_{i}$ − r$_{i}$ for i = 1 , . . . , n . In all discussions, the order related to e$_{i}$ , which is an error of the previously approx- imated zeros z$_{i}$ , is presumed to be the same. After that, we will show that the order related to e$_{i}$ , which is an error of the approximated zeros concerning each method, is identical. For the same being, the order related to the already approximated zeros ˆ e$_{i}$ is hypothesized to be identical as follows: In Section 2, we give a new method for finding simultaneously all simple zeros of poly- nomials constructed by applying the Weierstrass method to the x ∗ in the trapezoidal Newton’s method, and prove the convergence of the method. In Section 3, we present e$_{i}$ = O$_{M}$ ( e ) for all i.