Finding All Zeros

Pre-Calculus · Axiom Academy

Rational Root Theorem, synthetic division, and the quadratic formula on one cubic. Find all zeros (real and complex) of the polynomial f(x) = 2x^3 - 2x^2 - x - 6 . Nice work — you found every zero of a cubic by combining three core tools. Rational Root Theorem: possible rational zeros are — a finite list to test. Synthetic division: once a zero is found, dividing by its factor drops the degree, leaving a quadratic. Complex zeros come in conjugate pairs: the quadratic 2x^2 + 2x + 3 has a negative discriminant, so its zeros are complex conjugates. Result: the zeros are — three zeros for a degree-3 polynomial. Every polynomial of degree n has exactly n zeros over the complex numbers, counting multiplicity — so once the count matches the degree, you have them all.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.