Fundamental Theorem of Algebra
In short: The fundamental theorem of algebra states that every polynomial of degree n ≥ 1 has exactly n complex zeros when each zero is counted as often as its multiplicity.
PrecalculusAlgebraN-CN.C.9
What is Fundamental Theorem of Algebra?
The fundamental theorem of algebra promises that complex numbers are enough: a polynomial never leaves you without answers. A cubic has three zeros, a quartic has four, and so on, once you allow imaginary values and count repeats.
Formally, a polynomial \(f(x)\) of degree \(n \ge 1\) with complex coefficients can be factored as \(a(x – r_1)(x – r_2)\cdots(x – r_n)\) over the complex numbers. If the coefficients are real, nonreal zeros come in conjugate pairs, so \(a + bi\) is a zero only if \(a – bi\) is also a zero.
How to spot it
- A question asks how many zeros, real or complex, a polynomial must have, which the fundamental theorem of algebra answers.
- A polynomial is factored with a quadratic like x² + 9 that does not split over the real numbers.
- One complex zero such as 2 + i is given, and you must name the others for a real-coefficient polynomial.
- A graph crosses the x-axis fewer times than the degree, and the missing zeros must be explained.
The method
- Read the degree \(n\): by the fundamental theorem of algebra that is the total number of complex zeros, counting multiplicity.
- Find the real zeros and their multiplicities by factoring or from the graph.
- Subtract the count found from \(n\); the remaining zeros are nonreal.
- For real coefficients, pair each nonreal zero with its conjugate to account for them.
Fundamental Theorem of Algebra: solved example
Problem
How many complex zeros does \(f(x) = (x – 1)^2(x + 4)(x^2 + 25)\) have, and what are they?
Solution
Multiply the degrees of the pieces by their powers: \(2 + 1 + 2 = 5\), so there are 5 zeros counting multiplicity.
They are \(x = 1\) (twice), \(x = -4\), \(x = 5i\) and \(x = -5i\). That is five zeros: 1, 1, -4, 5i and -5i.
Try it
Question 1
How many complex zeros does \(7x^6 – x + 2\) have, counting multiplicity?
Show the answer
The degree is 6, so there are exactly 6.
Question 2
A cubic with real coefficients has the zero \(2 + i\). What can be said about the other zeros?
Show the answer
The conjugate \(2 – i\) is a zero too, and the third zero must be real.
Fundamental Theorem of Algebra vs. Rational Root Theorem
This statement tells you how many zeros exist, but it does not tell you what they are or how to find them. The rational root test is a practical tool that lists possible fraction zeros you can try. One guarantees the count; the other helps you hunt for actual values.
Keep going
Lessons
Worksheets and quizzes
- Complex Numbers and Complex Zeros – Worksheet
- Complex Numbers and Complex Zeros – Quiz
- Zeros and Graphs of Polynomial Functions – Worksheet
- Zeros and Graphs of Polynomial Functions – Quiz
Related words
- Rational Root Theorem
- Factor Theorem
- Complex Number
- Zeros of a Function
- Multiplicity of a Zero
- Complex Conjugate
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