Online Equation Solver

About solving equations

A value cc is said to be a root of a polynomial p(x)px if p(c)=0pc=0.

The largest exponent of xx appearing in p(x)px is called the degree of pp. If p(x)px has degree nn, then it is well known that there are nn roots, once one takes into account multiplicity. To understand what is meant by multiplicity, take, for example, 开始 幂, 开始 底数, x , 底数 结束,开始 指数, 2 , 指数 结束 , 幂 结束 - 6x + 9= 开始 幂, 开始 底数, (x-3) , 底数 结束,开始 指数, 2 , 指数 结束 , 幂 结束 =(x-3)(x-3)x2 - 6x + 9=x-32=x-3x-3. This polynomial is considered to have two roots, both equal to 3.

One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development.

Systems of linear equations are often solved using Gaussian elimination or related methods. This too is typically encountered in secondary or college math curricula. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools.