Finding Zeros with the Factor Theorem

Finding a polynomial's zeros from one known zero means dividing out its factor by synthetic division, then solving the smaller quotient — repeating until a quadratic remains for the old methods.

Find all zeros of f(x) = x³ − 3x + 2, given that −2 is a zero. The Factor Theorem converts the tip: x + 2 is a factor. Synthetic division by k = −2 on 1, 0, −3, 2 runs: 1, then −2, then 1, then 0 — quotient x² − 2x + 1, remainder 0 as promised. The quotient is a perfect square trinomial: (x − 1)². Full factorization: f(x) = (x + 2)(x − 1)², zeros −2 and 1, the second with multiplicity 2 — the same anatomy Video 360 sketched. Each division drops the degree by one; the machine always lands on a quadratic eventually, and quadratics always surrender.