In this section we show you how to work with one root or all roots root:symbolic of a polynomial. These roots are represented symbolically (as opposed to being numeric approximations). See ugxProblemOnePol and ugxProblemPolSys for information about solving for the roots of one or more polynomials.
Use rootOf to get a symbolic root of a polynomial: returns a root of .
This creates an algebraic number . algebraic number number:algebraic
To find the algebraic relation that defines , use definingPolynomial.
You can use in any further expression, including a nested rootOf.
Higher powers of the roots are automatically reduced during calculations.
The operation zeroOf is similar to rootOf, except that it may express the root using radicals in some cases. radical
Use rootsOf to get all symbolic roots of a polynomial: returns a list of all the roots of . If has a multiple root of order , then that root root:multiple appears times in the list. Make sure these variables are x0 etc.
Compute all the roots of .
As a side effect, the variables and are bound to the first three roots of .
Although they all satisfy and are different algebraic numbers. To find the algebraic relation that defines each of them, use definingPolynomial.
We can check that the sum and product of the roots of are its trace and norm.
Corresponding to the pair of operations rootOf/ zeroOf in ugxProblemOnePol , there is an operation zerosOf that, like rootsOf, computes all the roots of a given polynomial, but which expresses some of them in terms of radicals.
As you see, only one implicit algebraic number was created ( ), and its defining equation is this. The other three roots are expressed in radicals.