In this master’s thesis, we describe the historical development of polynomials and the Fundamental Theorem of Algebra, from Al-Khwarizmi’s Algebra to Galois theory. We continue the next chapter with the three classical Greek problems of mathematics, another popular historical problem. By introducing constructible numbers, we prove the unsolvability of the three classical problems using polynomials. In the next chapter, we study polynomials and their properties in detail. In particular, we look at zeros and factorization of polynomials following with Horner’s algorithm and examples how to efficiently divide a polynomial by the first degree polynomial. We study symmetric polynomials and state the Fundamental Theorem of Symmetric Polynomials as a key to proving the Fundamental Theorem of Algebra. Upon introducing a splitting field of a polynomial, we present an algebraic proof of the Fundamental Theorem of Algebra. At the end of the thesis, we show various problems related to polynomials that appeared at national, foreign and international competitions in mathematics.