Write a polynomial function P(x) with rational coefficients so that P(x)=0 has the given roots

Question

-5-7i and 2-\sqrt(11)

Answer 1

root are -5-7i and 2-sqr11

imaginary or irrational roots always come in conjugate pairs,

so there's 4 of them total

-5-7i, -5+7i, 2 -sqr11 and 2+sqr11

x = -5-7i

x +5 = -7i

x^2 +10x+ 25 =-49

x^2 +10x +74 is one factor

x= 2-sqr11

x -2 = -sqr11

x^2 -4x +4 = 11

x^2 -4x -7 is another factor

multiply them together

P(x) = (x^2+10x+74)(x^2-4x-7)

multiply it out and you get a 4th degree polinomial function

P(x) = x^4 +6x^3 +27x^2-366x - 518,

all terms with integer rational coefficients