Question

If (2-square root of 3 ) is a root of a polynomial with integer coefficients which of the follow must be another root

Answer 1

$\text{If k and l are the roots of a polynomial w(x), then}\ w(x)=a(x-k)(x-l).$

$(2-\sqrt3)-a\ root\ of\ a\ polynomial\\\\another\ root\ must\ be\ C.\ (2+\sqrt3),\ because\\\\\ [x-(2-\sqrt3)][x-(2+\sqrt3)]=(x-2+\sqrt3)(x-2-\sqrt3)\\\\=[(x-2)+\sqrt3][(x-2)-\sqrt3]\\\\\text{use}\ (a-b)(a+b)=a^2-b^2\\\\=(x-2)^2-(\sqrt3)^2\\\\\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\=x^2-2(x)(2)+2^2-3=x^2-4x+4-3=x^2-4x+1$

$Answer:\ \boxed{C.\ (2+\sqrt3)}$