Zero Finding Strategy

Algebra 2 · Axiom Academy

One repeatable procedure that finds every zero of a polynomial — rational, irrational, and complex — and tells you exactly when you are done. 1. The Whole Strategy at a Glance Every zero-finding problem runs through the same pipeline. Read it once as a decision procedure, then watch it solve x^3 - 6x^2 + 11x - 6 from start to finish — the active branch lights up as the polynomial travels through it. Factor out any common factor and any x^k — that peels off the zeros at x = 0 . If the degree is , factor or use the quadratic formula and stop. Otherwise list the rational candidates with the Rational Root Theorem. Test a candidate by synthetic division; a hit divides out, dropping the degree by one. Repeat on the smaller quotient until you reach a quadratic, then finish it. Count zeros with multiplicity — when the count equals the degree, you have them all. Our worked example for the whole lesson 2. First Move: Peel Off the Easy Zeros Before any theorem, factor out the greatest common factor and every power of x . A factor of x^k hands you the zero x = 0 immediately — repeated k times, which is what multiplicity means — and shrinks the degree so the rest of the work is smaller. Pull out the common factor 2x^2 The x^2 gives x = 0 twice — a double zero — and a quadratic is left over The factor x^2 is not one zero, it is x = 0 counted twice . Miss that and your final tally comes up short of the degree. The leftover x^2 - 2x - 3 = (x-3)(x+1) then gives x = 3 and x = -1 .

This is the written version of the interactive lesson above. See the full Algebra 2 course.