The topic of this thesis is to determine the position of the real roots of polynomials, i.e. to determine the bounds in which the real roots of polynomials might occur. In the first chapter we introduce the term polynomial and give some properties of polynomial divisibility. We define the highest common factor of the two polynomials and using Euclidean algorithm we prove that it is uniquely determined. We define the multiplicity of roots and show that the polynomial with real coefficients can have real roots and / or pairs of conjugated complex roots. We observe how the leading and constant term influence the value of the polynomial and can be used to determine the boundaries of the roots. We also give the proof of Bolzano’s theorem for polynomials needed to study the distribution of polynomial roots. In the second chapter we study three useful methods of separation the real roots of a polynomial. We describe a method, due to Sturm, for finding Sturm’s sequence of functions needed to determine the number of distinct real roots of a polynomial located in an interval. Using Fourier’s theorem we separate the real roots of a polynomial using only its derivatives. By applying the Descartes rule of signs, we determine positive and negative roots. In the last section of the second chapter, the proofs of Sturm’s and Fourier’s theorems are given. The paper presents a few examples illustrating the application of the aforementioned theorems and methods.