Find all real and complex zeros of a polynomial using the Rational Root Theorem, synthetic division, and the quadratic formula. Works for polynomials up to degree 4.
Step 1 — Rational Root Theorem: If p(x) has integer coefficients, every rational zero has the form ±(factor of constant term) / (factor of leading coefficient).
Step 2 — Test candidates: Substitute each candidate k into p(k). If p(k) = 0, then k is a zero and (x − k) is a factor.
Step 3 — Synthetic division: Divide p(x) by (x − k) to get a reduced quotient polynomial of degree n − 1.
Step 4 — Repeat: Apply RRT and synthetic division again on the reduced polynomial until you reach a quadratic or linear factor.
Step 5 — Quadratic formula: Solve any remaining quadratic factor ax² + bx + c = 0 using the quadratic formula. If Δ < 0, the roots are complex.
deg(p) = n → exactly n zeros (in ℂ, counting multiplicity)
Every non-constant polynomial with complex coefficients has at least one complex zero. This means a degree n polynomial has exactly n zeros when you count multiplicity and include complex (imaginary) roots.
Consequences for real-coefficient polynomials:
One-on-one Algebra 2 tutoring builds real fluency with the Rational Root Theorem, synthetic division, and complex roots — so the algorithm becomes second nature, not a memorized recipe.