Find the roots of the polynomial equation
x^4 + 16x^2 = 0
To find the roots of the polynomial equation x^4 + 16x^2 = 0, we can set the equation to equal zero and factor out an x^2 term:
x^4 + 16x^2 = 0
x^2(x^2 + 16) = 0
This equation can be true if either x^2 = 0 or x^2 + 16 = 0.
1) If x^2 = 0, then x = 0 is a root of the equation.
2) If x^2 + 16 = 0, then we can subtract 16 from both sides of the equation:
x^2 + 16 = 0
x^2 = -16
Since the square of a real number cannot be negative, there are no real roots for this equation. However, we can factor out -1:
x^2 = -16
x^2 + 16 = 0
(x + 4i)(x - 4i) = 0
Therefore, the two complex roots of the equation are x = 4i and x = -4i.