+16 Find a monic quartic polynomial \(f(x)\) with rational coefficients whose roots include \(x = 1 -sqrt(2)\) and \(x = 2+sqrt(5)\). Give your answer in expanded form.
+23252 If 1 - sqrt(2) is a solution, so is 1 + sqrt(2).
If 2 + sqrt(5) is a solution, so is 2 - sqrt(5).
The factors will be: [ x - ( 1 - sqrt(2) ) ] · [ x - ( 1 + sqrt(2) ) ] · [ x - ( 2 + sqrt(5) ) ] · [ x - ( 2 - sqrt(5) ) ]
Multiplying out [ x - ( 1 - sqrt(2) ) ] · [ x - ( 1 + sqrt(2) ) ] ---> [ x2 - 2x - 1 ]
Multiplying out [ x - ( 2 + sqrt(5) ) ] · [ x - ( 2 - sqrt(5) ) ] ---> [ x2 - 4x - 1 ]
Multiplying out [ x2 - 2x - 1 ] · [ x2 - 4x - 1 ] ---> x4 - 6x3 + 6x2 + 6x + 1
