Exam tip: Always scan the question prompt for wording about "unique" or "real" zeros. If no qualifier is given, "number of zeros" by convention means total zeros counting multiplicity per the FTA on the AP exam.
2. 复共轭根定理★★★☆☆⏱ 4 min
This is the most commonly tested result for this topic on the AP exam, and it applies to all polynomials you will encounter on the test. It allows you to immediately get a second zero when given one non-real complex zero, so you can form a quadratic factor with real coefficients and avoid complex division.
Exam tip: You can use the shortcut $x^2 - 2ax + (a^2 + b^2)$ for the quadratic factor for roots $a \pm bi$ to skip expanding the product every time, which saves time on both MCQ and FRQ.
3. Factoring Over Reals vs. Complex Numbers★★★☆☆⏱ 4 min
When factoring over the real numbers, non-real complex zeros produce linear factors with complex coefficients, which are not allowed. Instead, we pair conjugate non-real zeros into irreducible quadratic factors with real coefficients (irreducible means they cannot be factored further into linear terms with real coefficients). A key result: the number of non-real complex zeros (counting multiplicity) is always even, so the number of real zeros has the same parity as the degree of the polynomial. All odd-degree polynomials with real coefficients therefore have at least one real zero.
Exam tip: If a question asks for factoring over the reals, do not split the irreducible quadratic into linear factors with complex coefficients — this will cost you points on FRQs.
4. AP-Style Concept Check★★★★☆⏱ 3 min
Common Pitfalls
Why: Confuses the rule that non-real complex zeros come in pairs with the FTA's total zero count rule. Students forget pairing only applies to non-real zeros, not all complex zeros.
Why: Students confuse flipping the sign of the real part instead of the imaginary part when finding conjugates.
Why: Assumes "number of zeros" always means unique zeros.
Why: Forgets non-real complex zeros come in pairs, so the number of real zeros must have the same parity as the degree.
Why: Forgets that non-real zeros have non-real coefficients in their linear factors, which are not allowed when factoring over the reals.