Using All Roots of a Polynomial
Use rootsOf to get all symbolic roots
of a polynomial. The call rootsOf(p,x) returns a list of all the roots
of p(x). If p(x) has a multiple root of order n, then that root appears
n times in the list.
Compute all the roots of x^4+1.
As a side effect, the variables %x0, %x1, and %x2 are bound to the first
three roots of x^4+1.
Although they all satisfy x^4+1=0, %x0, %x1, and %x2 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 x^4+1 are its
trace and norm.
Corresponding to the pair of operations
rootOf and
zeroOf in
Solution of a Single Polynomial Equation
there is an operations 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 (%y1), and its
defining equation is this. The other three roots are expressed in radicals.
