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Question

Form the equation of the fourth degree with rational coefficients, one of whose roots is 2+3.

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Solution

Here we must have 2+3,23 as one pair of roots, and 2+3,23 as another pair.
Corresponding to the first pair we have the quadratic factor x222x+5, and corresponding to the second pair we have the quadratic factor
x2+22x+5.
Thus the required equation is
(x2+22x+5)(x222x+5)=0,
or (x2+5)28x2=0,
or x4+2x2+25=0.

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