How do you find all additional roots given the roots $3-sqrt2$ and $1+sqrt3$ ?

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Answer 1 · 2016-10-03T15:56:31

$3+sqrt 2 and 1 - sqrt 3$

If the coefficients of a polynomial equation are rational, complex

roots and irrational roots occur in conjugate pairs.

For examples.

the roots of $x^3-x^4-8x^3+8x^2+15x-15=0$ are

$+-sqrt 3, +-sqrt5 and 1$ .

The roots of $x^3+x^2+x+1=0$ are $+-i and -1$ .

Here the conjugates 3+sqrt 2 and 1 - sqrt 3 should also be the roots

of such an equation.

In the case of surd like $sqrt(a+sqrtb)$ , the additional roots will be

$sqrt(a-sqrtb), -sqrt(a+sqrtb) and -sqrt(a-sqrtb).$ Here, altogether it

is a double couple $+-sqrt(a+-sqrtb)$ ..,

How do you find all additional roots given the roots 3-sqrt23-sqrt2 and 1+sqrt31+sqrt3?