Section8.3

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## 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

a 4+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

(10 a 3+11 a 2+2 a-4)b+15 a 3+10 a 2+4 a-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)

-3-1 2

Type: Expression Integer
rootOf(e**5-2,e)

e

Type: Expression Integer
zeroOf(f**5-2,f)

2 5

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

%x0 4+1

Type: Expression Integer
definingPolynomial %x1

%x1 2+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%x0 2

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+1 2,-1-1 2,--1-1 2,--1+1 2]

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

%%var 2+1

Type: Expression Integer
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