Fundamental Theorem of Algebra
Every degree-n polynomial has exactly n complex roots counted with multiplicity.
Key idea
Every degree-n polynomial has exactly n complex roots counted with multiplicity.
In depth
Every non-constant polynomial of degree n has exactly n complex roots, counting multiplicity. This guarantees that polynomials always factor completely over the complex numbers.
Worked example
x³ − 1 = 0 has roots 1 and the two complex numbers (−1 ± i√3)/2.
Real-world uses
- Knowing how many solutions a polynomial control or signal model can have
Common mistake
Assuming all roots are real.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Fundamental Theorem of Algebra?
Every degree-n polynomial has exactly n complex roots counted with multiplicity. It is a topic at the expert level in Algebra.
Where is Fundamental Theorem of Algebra used in real life?
Real-world examples: Knowing how many solutions a polynomial control or signal model can have.
What should I learn before Fundamental Theorem of Algebra?
Make sure you are comfortable with Rational Root Theorem, the previous topic in Algebra.
What is a common mistake with Fundamental Theorem of Algebra?
Assuming all roots are real.
How can I practise Fundamental Theorem of Algebra?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Fundamental Theorem of Algebra?
Move on to Complex Numbers next.
Related expert topics in Algebra
- Linear Programming – Optimise a linear objective over a feasible polygon; maxima occur at vertices.
- Rational Root Theorem – Possible rational roots are ± factors of constant over factors of leading coefficient.
- Binomial Theorem – Expand (a+b)ⁿ with binomial coefficients.
- Partial Fractions – Decompose a rational function into simpler fractions.
- Mathematical Induction – Prove statements for all n by base case and inductive step.
- Cramer’s Rule – Solve linear systems using determinants.