The method of uncertain coefficients in the course of algebra

The tasks and problems reflecting universality and beauty of the method of uncertain coefficients in a school course of algebra are considered in the article. The idea of the method will allow pupils to expand the ideas of operation with polynomials, to reveal another way of division of a polynomial into a polynomial, to acquire skills of disposal of irrationality in a fraction denominator, to learn to display the proper rational fraction into the elementary ones. The examples given in the article can be useful for both mathematics teachers, and the pupils who are seriously involved in mathematics.