Key takeaway: The fundamental theorem of algebra states that every nonconstant polynomial of degree n has exactly n complex roots when repeated roots are counted with multiplicity.
Grade 12 – PrecalculusN-CN.C.9Expressions and Equations
Definition of Fundamental Theorem of Algebra
The fundamental theorem of algebra guarantees that solving a polynomial equation never leaves you empty-handed, provided complex numbers are allowed. A degree 5 polynomial has five roots, a degree 12 polynomial has twelve, and so on. Some may repeat, and some may be non-real.
Non-real roots of a polynomial with real coefficients always come in conjugate pairs, such as \(2i\) and \(-2i\). So an odd-degree real polynomial always has at least one real root, while even-degree ones can have none. Together with the factor theorem, the result says every polynomial splits completely into linear factors over the complex numbers.
Formula and notation
For a polynomial of degree \(n\ge 1\) with leading coefficient \(a\):
\[P(x)=a\,(x-r_1)(x-r_2)\cdots(x-r_n),\]
where each \(r_i\) is a complex number, not necessarily different from the others. A root that appears \(m\) times has multiplicity \(m\).
Key facts
- The count of roots equals the degree, including repeated and non-real ones.
- A real graph shows only real roots, as x-intercepts.
- For real coefficients, the conjugate of a non-real root is also a root.
- A polynomial of odd degree must cross the x-axis at least once.
Fundamental Theorem of Algebra example
Find all the roots of \(P(x)=x^3-3x^2+4x-12\) and confirm the total matches the degree.
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Pair the first two terms and the last two. Both pairs share the binomial \(x-3\), so the cubic factors into that binomial times \(x^2+4\).
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The first factor vanishes at 3.
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The second needs \(x^2=-4\), so the square roots of \(-4\) are \(2i\) and \(-2i\).
The roots are 3, 2i and −2i: three roots for a degree 3 polynomial.
Common mistake
Watch out
Callie sees the graph touch the x-axis once and concludes that the cubic has only one root.
Why it is wrong
A graph shows only real roots. The cubic still has three roots by the fundamental theorem of algebra; the other two, \(2i\) and \(-2i\), are non-real and cannot be seen on a real-number plane.
Related terms and lessons
- Factor Theorem
- Remainder Theorem
- Rational Root Theorem
- Polar Form of a Complex Number
- De Moivre's Theorem
- Precalculus · Polynomial and Rational Functions in DepthLessonWorksheetQuiz
- Precalculus · Complex Factorization and Volume of a SphereLessonWorksheetQuiz
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