The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every polynomial equation of degree \(n\) with complex number coefficients has \(n\) roots, or solutions, in the complex numbers.

The Fundamental Theorem of Algebra is an example of an ‘existence’ theorem in mathematics, like the Intermediate Value Theorem. This theorem guarantees the existence of at least one zero but gives us no algorithm to find it.
Step by step guide to the fundamental theorem of algebra
Every polynomial function contains at least one complex zero, according to the Fundamental Theorem of Algebra. The cornerstone for solving polynomial equations is this theorem. There is at least one complex root in every polynomial equation with complex coefficients and degrees. Gauss was the first to prove this theorem. It’s the same as saying that a polynomial \(P(z)\) of degree \(n\) has \(n\) values \(z_i\) (some of them possibly degenerate) for which \(P(z_i)=0\). Polynomial roots are the names given to such numbers.
An example of a polynomial with a single root of multiplicity \(>1\) is \(z^2-2z+1=(z-1)(z-1)\) which has \(z=1\) as a root of multiplicity \(2\).
The Fundamental Theorem of Algebra – Example 1:
what are the roots of \(x^2-4\)?
\(x^2-4\) has a degree of \(2\) (the largest exponent of \(x\) is \(2\)), so there are 2 roots.
Let us solve it. We want it to be equal to zero:
\(x^2-4=0\)
\(x^2=4\)
Now, take the square root of both sides:
\(x=\pm2\)
So, the roots are \(+2\) and \(-2\).
Exercises for the Fundamental Theorem of Algebra
- Write the polynomial that has the following roots \(4\) (with multiplicity \(3\)), \(2\) (with multiplicity \(2\)) and \(0\).
- Find the zeroes of the polynomial \(x^2+3x+2\).
- Find the zeroes of the polynomial \(x^3-4x^2-4x+16\).
- How many roots does the following equation have? \(\left(2x^2+2x-1\right)\left(3x^2-9\right)=0\)

- \(\color{blue}{f(x)=(x-4)^3(x-2)^2 x}\)
- \(\color{blue}{x=-1, \:or\: x=-2}\)
- \(\color{blue}{x=2, x=-2 \:or\: x=4}\)
- \(\color{blue}{4}\)
Related to This Article
More math articles
- ACT Calculators: Exреrt Tiрѕ
- 5 Best Laptops For Teachers
- 10 Most Common AFOQT Math Questions
- Number Properties Puzzle – Challenge 16
- Full-Length 8th Grade FSA Math Practice Test
- 3rd Grade WVGSA Math Worksheets: FREE & Printable
- Full-Length SAT Math Practice Test
- Full-Length 6th Grade ACT Aspire Math Practice Test-Answers and Explanations
- 5th Grade PEAKS Math Worksheets: FREE & Printable
- How to do well on the PSAT test?
What people say about "The Fundamental Theorem of Algebra"?
No one replied yet.