The simplest method is based on the intermediate value theorem and repeatedly divides and interval in half to see where the function changes sign.
One computational robust algorithm for doing this has no shift, fixed shift, and variable shift stages. For complex values, the Lehmer-Schur algorithm does this. In addition to the Jenkins-Traub method, a common false position approach is Ridder's algorithm. Using Horner scheme to quickly evaluate a polynomial via synthetic division and Rufini's algorithm to deflate the polynomial is helpful in this task. Laguerre's method can do this for polynomial, but in general this can be achieved by a (*) fixed point method where the function and its derivative are subtracted from the function variable. The simplest method is based on the intermediate value theorem and repeatedly divides and interval in half to see where the function changes sign. For 10 points, name this task that can be achieved by bisection method and Newton's method and involves finding where a function is equal to zero.
One computational robust algorithm for doing this has no shift, fixed shift, and variable shift stages.
root finding [accept solving for a root of a function; accept finding a zero of a function or word forms before mention; accept word forms; accept accelerating the convergence of a sequence before Horner scheme, prompt after; prompt on anything involving a fixed point]