Question

State how many imaginary and real zeros the function has.
f(x) = x^4 + 12 x^3 + 37x^2 + 12x + 36

Answer 1

$x^4+12x^3+37x^2+12x+36=0\\\\x^4+12x^3+36x^2+x^2+12x+36=0\\\\ x^2(x^2+12x+36)+(x^2+12x+36)=0\\\\x^2(x+6)^2+(x+6)^2=0\\\\(x^2+1)(x+6)^2=0\\\\\\x^2+1=0\qquad\vee\qquad x+6=0\qquad\vee\qquad x+6=0\\\\x^2=-1\qquad\vee\qquad x=-6\qquad\vee\qquad x=-6\\\\\boxed{x=i\qquad\vee\qquad x=-i\qquad\vee\qquad x=6\qquad\vee\qquad x=6}$

The function has two imaginary and two real zeros.