To solve the equation [tex]\( x^3 - 3x^2 - 4x + 12 = 0 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that make the equation true. These values are also known as the roots of the equation. Here's a step-by-step breakdown of the solution:
1. Identify the equation:
We start with the polynomial equation:
[tex]\[
x^3 - 3x^2 - 4x + 12 = 0.
\][/tex]
2. Factorization (If applicable):
While this cubic equation can be approached by factorization, it is often more efficient to use algebraic techniques to find the roots directly.
3. Finding the roots:
In this case, solving the equation, we find that there are three real roots. These roots can be confirmed through algebraic methods such as synthetic division, the Rational Root Theorem, or numerical solvers.
4. Verification of roots:
By verifying through the algebraic process or numerical solutions, we determine the roots of the equation to be:
[tex]\[
x = -2, \quad x = 2, \quad x = 3.
\][/tex]
5. Summarizing the solution:
Therefore, the solutions to the equation [tex]\( x^3 - 3x^2 - 4x + 12 = 0 \)[/tex] are:
[tex]\[
x = -2, \quad x = 2, \quad x = 3.
\][/tex]
These values are the roots of the polynomial equation, meaning they satisfy the equation when substituted back in for [tex]\( x \)[/tex].
