Question

How do you solve this? (#11/#12)

Answer 1

$\text{If}\ x_1,\ \text{and}\ x_2\ \text{are zeros of a polynomial function, then the function}\\\text{has equation}:\\\\f(x)=a(x-x_1)(x-x_2).\\----------------------------\\\\11.\ \text{We have the zeros}\ 3\ \text{and}\ i:\\\\f(x)=(x-3)(x-i)=(x)(x)+(x)(-i)+(-3)(x)+(-3)(-i)\\\\=x^2-xi-3x+3i=\boxed{x^2-(3+i)x+3i}\\\\12.\ \text{We have the zeros}\ -2\ \text{and}\ 2i:\\\\f(x)=(x-(-2))(x-2i)=(x+2)(x-2i)\\\\=x^2-(2i)^2=x^2-2^2i^2=x^2-4(-1)=\boxed{x^2+4}\\\\\text{used}\\\\(a-b)(a+b)=a^2-b^2\\\\i=\sqrt{-1}\to i^2=-1$