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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.
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Your monic quartic polynomial is p(x) = (x-(1-sqrt(2))*(x-(1+sqrt(2))*(x-(2+sqrt(5))*(x-(2-sqrt(5)).
It is simply the product of four binomials assosiated with the GIVEN roots and with theis conjugate roots
over rational numbers.
The product of the first and the second binomials is
(x-(1-sqrt(2))*(x-(1+sqrt(2)) = ((x-1)-sqrt(2))*((x-1)+sqrt(2)) = (x-1)^2 -4 = x^2 - 2x + 1 - 4 = x^2 - 2x -3.
The product of the third and the fourth binomials is
(x-(1-sqrt(5))*(x-(1+sqrt(5)) = ((x-1)-sqrt(5))*((x-1)+sqrt(5)) = (x-1)^2 -25 = x^2 - 2x + 1 - 25 = x^2 - 2x -24.
The monic quartic polynomial is p(x) = (x^2 - 2x -3)*(x^2 - 2x -24) = x^4 - 4x^3 -23x^2 + 54x + 72.
It is your answer.
Solved.
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