Find a monic quartic polynomial with rational coefficients whose roots include 2+sqrt(2) and 1-sqrt(5) . Give your answer in expanded form.
+9519 The coefficients are rational => Sum of roots and product of roots are also rational.
=> conjugates of 2 + sqrt 2 and 1 - sqrt 5 must also be roots of the polynomial.
=> The roots are 2 + sqrt 2, 2 - sqrt 2, 1 - sqrt 5, 1 + sqrt 5.
The required monic quartic polynomial is
\(\quad(x - (2 + \sqrt 2))(x - (2 - \sqrt 2))(x - (1 - \sqrt 5))(x - (1 + \sqrt 5))\\ =(x^2 - 4x + 2)(x^2 - 2x - 4)\\ =x^4 - 6x^3 +6x^2 +12x-8\)
+9519 The coefficients are rational => Sum of roots and product of roots are also rational.
=> conjugates of 2 + sqrt 2 and 1 - sqrt 5 must also be roots of the polynomial.
=> The roots are 2 + sqrt 2, 2 - sqrt 2, 1 - sqrt 5, 1 + sqrt 5.
The required monic quartic polynomial is
\(\quad(x - (2 + \sqrt 2))(x - (2 - \sqrt 2))(x - (1 - \sqrt 5))(x - (1 + \sqrt 5))\\ =(x^2 - 4x + 2)(x^2 - 2x - 4)\\ =x^4 - 6x^3 +6x^2 +12x-8\)
