Find all real and complex zeros of a polynomial up to degree 5. Enter coefficients, then see every step: rational root candidates, synthetic division tables, quadratic formula for irreducible quadratics, and a graph with all real zeros marked.
If p(x) = aₙxⁿ + … + a₀ (integer coefficients),
every rational zero has form ±p/q where
p | a₀ (factor of constant term)
q | aₙ (factor of leading coefficient)
This theorem drastically limits the candidates to test. Once you list ±(factors of a₀)÷(factors of aₙ), you check each by evaluating p(candidate) — if it equals 0, you have a zero.
After finding a zero r, perform synthetic division by (x − r) to get a reduced polynomial of degree n − 1. Repeat until you reach a quadratic.
deg(p) = n ⟹ exactly n zeros in ℂ
(counting multiplicity)
Every polynomial of degree n has exactly n zeros when you count over the complex numbers ℂ and include repeated roots with multiplicity. This guarantees you will always find the right number of zeros.
One-on-one Precalculus tutoring builds the fluency to move through the Rational Root Theorem, synthetic division, and complex roots quickly and accurately — turning a multi-step algorithm into a reliable routine.