Numerical Solution of Polynomial Equations

When (RefLink[Solve, paclet:ref/Solve]) cannot find solutions in terms of radicals to polynomial equations, it returns a symbolic form of the result in terms of (RefLink[Root, paclet:ref/Root]) objects:

Solve[x^5+7x+1==0,x]

You can get numerical solutions by applying (RefLink[N, paclet:ref/N]):

N[%]

This gives the numerical solutions to 25-digit precision:

N[%%,25]

You can use (RefLink[NSolve, paclet:ref/NSolve]) to get numerical solutions to polynomial equations directly, without first trying to find exact results:

NSolve[x^7+x+1==0,x]

Numerical solution of polynomial equations and systems.

(RefLink[NSolve, paclet:ref/NSolve]) will give you the complete set of numerical solutions to any polynomial equation or system of polynomial equations.

(RefLink[NSolve, paclet:ref/NSolve]) can find solutions to sets of simultaneous polynomial equations:

NSolve[{x^2+y^2==1,x^3+y^3==2},{x,y}]