The fundamental theorem of algebra & conjugate roots
The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients has at least one complex root — and by induction, exactly n roots counted with multiplicity. The conjugate root theorem adds that when the coefficients are REAL, non-real roots come in conjugate pairs, which forces every real polynomial to factor into real linear and quadratic pieces.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Count the roots by degreeA degree-n polynomial has exactly n complex roots with multiplicity. No more, no fewer.
- Check whether the coefficients are realThe conjugate pairing applies only then, and it is the hypothesis most often overlooked.
- Pair up the non-real rootsEach pair a ± bi contributes an irreducible real quadratic factor.
- Deduce structural factsAn odd-degree real polynomial has at least one real root, since the non-real ones pair off evenly.
Worked example
A real quartic has roots 1 + 2i and 3. What is the fourth root, and what is the polynomial?
- The coefficients are real, so 1 − 2i is also a root.
- That gives three roots: 1 + 2i, 1 − 2i and 3. A quartic has four.
- The conjugate pair is already complete, so the fourth root must be real — say r.
- The complex pair gives the factor (z−1)² + 4 = z² − 2z + 5.
Answer. The fourth root is real (undetermined without more information). The polynomial is (z² − 2z + 5)(z − 3)(z − r) for some real r.
Where marks get dropped
These are the specific errors that cost credit on the fundamental theorem of algebra & conjugate roots questions — QED's rubric penalises each of them separately.
- Applying conjugate pairing to polynomials with complex coefficients, where it simply fails.
- Counting distinct roots rather than roots with multiplicity. (z−1)³ has one distinct root but degree 3.
- Assuming a real polynomial factors into real LINEAR factors. Irreducible real quadratics such as z² + 1 cannot be split further over ℝ.
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The fundamental theorem of algebra & conjugate roots — frequently asked questions
Why is it called "fundamental"?
Because it says ℂ is algebraically closed — no further extension is needed to solve polynomial equations. Extending ℝ to ℂ finishes the job completely.
Does the theorem tell me how to find the roots?
No, it is purely an existence result. Indeed by Abel–Ruffini there is no general radical formula beyond degree 4, so numerical methods are needed.
What does it say about real factorisation?
Every real polynomial factors into real linear factors and irreducible real quadratics, because the non-real roots pair into quadratics with real coefficients.
The rest of Complex Numbers
Arithmetic, the Argand plane, polar form and De Moivre. Each subtopic below has its own method, worked example and mark-losing traps.
- 1The imaginary unit i & complex arithmetic
- 2Conjugates & division
- 3Modulus, argument & the Argand diagram
- 4Polar & exponential form re^{iθ}
- 5Multiplying & dividing in polar form
- 6De Moivre’s theorem & powers
- 7nth roots & roots of unity
- 8Solving polynomial equations over ℂ
- 9The fundamental theorem of algebra & conjugate roots
- 10Euler’s formula & trigonometric identities
- 11Loci & regions in the complex plane
- 12Complex multiplication as rotation & scaling
- 13Applications: phasors & AC circuits
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