8.3 Manipulating Symbolic Roots of a Polynomial

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.

8.3.1 Using a Single Root of a Polynomial

Use rootOf to get a symbolic root of a polynomial: rootOf(p,x) returns a root of p(x).

This creates an algebraic number a. algebraic number number:algebraic

a := rootOf(a^4+1,a)

\[\]

a

_{Type: Expression Integer}

To find the algebraic relation that defines a, use definingPolynomial.

definingPolynomial a

\[\]

a4+1

_{Type: Expression Integer}

You can use a in any further expression, including a nested rootOf.

b := rootOf(b^2-a-1,b)

\[\]

b

_{Type: Expression Integer}

Higher powers of the roots are automatically reduced during calculations.

a + b

\[\]

b+a

_{Type: Expression Integer}

% ^ 5

\[\]

(10a3+11a2+2a-4)b+15a3+10a2+4a-10

_{Type: Expression Integer}

The operation zeroOf is similar to rootOf, except that it may express the root using radicals in some cases. radical

rootOf(c^2+c+1,c)

\[\]

c

_{Type: Expression Integer}

zeroOf(d^2+d+1,d)

\[\]

-3-12

_{Type: Expression Integer}

rootOf(e^5-2,e)

\[\]

e

_{Type: Expression Integer}

zeroOf(f^5-2,f)

\[\]

25

_{Type: Expression Integer}

8.3.2 Using All Roots of a Polynomial

Use rootsOf to get all symbolic roots of a polynomial: 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 root:multiple appears n times in the list. Make sure these variables are x0 etc.

Compute all the roots of x**4+1.

l := rootsOf(x^4+1,x)

\[\]

[%x0,%x0%x1,-%x0,-%x0%x1]

_{Type: List Expression Integer}

As a side effect, the variables %x0,%x1 and %x2 are bound to the first three roots of x**4+1.

%x0^5

\[\]

-%x0

_{Type: Expression Integer}

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.

definingPolynomial %x0

\[\]

%x04+1

_{Type: Expression Integer}

definingPolynomial %x1

\[\]

%x12+1

_{Type: Expression Integer}

definingPolynomial %x2

\[\]

-%x2+%%var

_{Type: Expression Integer}

We can check that the sum and product of the roots of x**4+1 are its trace and norm.

x3 := last l

\[\]

-%x0%x1

_{Type: Expression Integer}

%x0 + %x1 + %x2 + x3

\[\]

(-%x0+1)%x1+%x0+%x2

_{Type: Expression Integer}

%x0 * %x1 * %x2 * x3

\[\]

%x2%x02

_{Type: Expression Integer}

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.

zerosOf(y^4+1,y)

\[\]

[-1+12,-1-12,–1-12,–1+12]

_{Type: List Expression Integer}

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.

definingPolynomial %y1

\[\]

%%var2+1

_{Type: Expression Integer}