How do you find all additional roots given the roots 3-sqrt232 and 1+sqrt31+3?

1 Answer
Oct 3, 2016

3+sqrt 2 and 1 - sqrt 33+2and13

Explanation:

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=0x3x48x3+8x2+15x15=0 are

+-sqrt 3, +-sqrt5 and 1±3,±5and1.

The roots of x^3+x^2+x+1=0x3+x2+x+1=0 are +-i and -1±iand1.

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)a+b, the additional roots will be

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

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