## Results on Resultants |

**Professor Pedro Teixeira**

Union College

January 24, 2005

4:30 pm

Blue House

Tea and pastries will be served

While it's well known that a polynomial equationf(x)=0 with complex coefficients always has a solution (Fundamental Theorem of Algebra), finding the solution(s) is often a very challenging problem. Equations of degree 2 are handled by the quadratic formula, and formulas of a similar flavor (but a lot "messier") take care of equations of degrees 3 and 4. However, for degree 5 and higher no similar formulas exist. In view of this fact, it's surprising that given any two polynomial equationsf(x)=0 andg(x)=0, it's always possible to determine whether the equations have a common solution, just using the coefficients offandgand the arithmetic operations... and it isn't hard either! One possible method is to use theEuclidean Algorithmto find the GCD offandg. In this talk we'll focus on a second method, which is by usingresultants. We'll see how resultants can also be used to determine whether a polynomial has multiple roots, to "solve" parametric equations, and to help solving systems of polynomial equations in two variables, and how, unlike the Euclidean Algorithm, the resultant generalizes in many directions.

For additional information, send e-mail to math@union.edu or call (518) 388-6246.

Union College Math Department Home PageComments to: math@union.edu Created automatically on: Sun Jan 21 22:57:30 EST 2018 |