Question

The product of the roots of the equation \( x^4 + 1 = 0 \) is ....... (answer in integer).

Hint:

For equations of the form \( x^n + a = 0 \), the product of the roots can be determined using the relationship for the product of the roots in polynomials.

Answer 1

The equation is given as \( x^4 + 1 = 0 \), which simplifies to: \[ x^4 = -1 \] The fourth roots of \( -1 \) are the complex numbers that satisfy this equation. These roots are given by: \[ x = e^{i\frac{(2n+1)\pi}{4}} \quad {for} \quad n = 0, 1, 2, 3 \] The four roots are: \[ x = e^{i\frac{\pi}{4}}, \, e^{i\frac{3\pi}{4}}, \, e^{i\frac{5\pi}{4}}, \, e^{i\frac{7\pi}{4}} \] The product of these roots can be found by multiplying all the roots. Since the product of the roots of a polynomial \( ax^n + bx^{n-1} + \cdots + z = 0 \) is given by \( (-1)^n \times \frac{{constant term}}{{leading coefficient}} \), for this equation, the product of the roots is \( (-1)^4 \times \frac{1}{1} = 1 \).
Thus, the product of the roots is \( \boxed{1} \).