Similarly, cubic equations with three real solutions have an algebraic solution that is unhelpful in that it contains cube roots of complex numbers; again an alternative solution exists in terms of trigonometric functions of real terms.
32.
In the case of the standard annuity formula, however, there is no closed-form algebraic solution for the interest rate ( although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms ).
33.
The Abel-Ruffini theorem states that the general quintic equation lacks an algebraic solution, and this directly implies that the general polynomial equation of degree " n ", for " n " e " 5, cannot be solved algebraically.
34.
I know you guys don't do homework, but I've got this question and our teacher just told us to use trial and error or the Pythagorean triplets to solve it because he said that the algebraic solution is too difficult.
35.
Permutations were studied by Joseph-Louis Lagrange in his 1770 paper " R�flexions sur la r�solution alg�brique des �quations ( Thoughts on the algebraic solution of equations ) " devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents.
36.
Then a necessary condition is that for almost all prime numbers " p ", the system defined by reduction modulo " p " should also have a full set of algebraic solutions, over the finite field with " p " elements.
37.
The study of such algebraic solutions involves examining the topology of the deformation parameter space ( and in particular, its mapping class group ); for the case of simple poles, this amounts to the study of the action of braid groups.
38.
:: : As a general principle, one of the goals in algebraic solutions to problems is to change the problem to be either a linear equation or a quadratic ( which can be effectively treated as a pair of linear equations ).
39.
In differential Galois theory the case of algebraic solutions is that in which the differential Galois group " G " is finite ( equivalently, of dimension 0, or of a finite monodromy group for the case of Riemann surfaces and linear equations ).
40.
:: ( a ) " Algebraic solution for geometry from dimensional constraints ", J . C . Owen, Proceedings of the first ACM symposium on Solid modeling foundations and CAD / CAM applications, p . 397-407, June 05-07, 1991, Austin, Texas, United States
