Question

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

Hint:

Remember: product of roots for even-degree polynomials is always positive if both leading and constant coefficients are positive.

Answer 1

Step 1: General formula for product of roots.
For a polynomial of degree \(n\): \[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0, \] the product of roots = \((-1)^n \frac{a_0}{a_n}\). Step 2: Identify coefficients.
Equation: \(x^4 + 1 = 0\). Here, \(a_n = 1, \; a_0 = 1, \; n=4\). Step 3: Substitution.
\[ \text{Product of roots} = (-1)^4 \cdot \frac{1}{1} = 1. \] Final Answer: \[ \boxed{1} \]