Use the Rational Root Theorem to find rational zeros of a polynomial, and apply the Fundamental Theorem of Algebra and the Complex Conjugate Theorem to find every zero, real or complex.
The Rational Root Theorem narrows an overwhelming search down to a short list: for a polynomial with integer coefficients, any RATIONAL zero must equal a factor of the constant term divided by a factor of the leading coefficient. Test each candidate (synthetic division is perfect for this) until one gives a remainder of 0, then divide it out and repeat on the smaller quotient.
The Fundamental Theorem of Algebra guarantees that a degree-n polynomial has exactly n zeros once you count complex zeros and repeated zeros properly -- nothing is ever missing. Complex (non-real) zeros of a polynomial with REAL coefficients always come in conjugate pairs: if a+bi is a zero, so is a-bi. That pairing is why a cubic with real coefficients either has three real zeros, or exactly one real zero and one conjugate pair -- never two complex zeros alone.
Mastery looks like: They generate a correct candidate list, find every rational zero by testing and dividing, and correctly identify any remaining zeros as a conjugate pair when the quotient is irreducible over the reals.
Common stumbles: Forgetting to divide out a confirmed zero before continuing the hunt; assuming a lone complex zero is possible for a real-coefficient polynomial.