Alright, let's solve the function step-by-step:
We are given the cubic equation:
[tex]\[ x^3 + 5x^2 - 3x - 15 = 0 \][/tex]
To find the roots of this equation, we determine the values of [tex]\( x \)[/tex] that satisfy it. Here are the steps taken to solve it, presented in detail:
1. Identify the cubic equation:
[tex]\[ x^3 + 5x^2 - 3x - 15 = 0 \][/tex]
2. Factorization:
For cubic equations, we usually check for rational roots using the Rational Root Theorem. However, solving cubic equations by factorization often involves trying possible roots, synthetic division, or applying formulas for cubic equations.
3. Find the Roots:
After solving the equation, the roots are found to be:
[tex]\[ x = -5, \quad x = -\sqrt{3}, \quad \text{and} \quad x = \sqrt{3} \][/tex]
Thus, the solutions to the equation [tex]\( x^3 + 5x^2 - 3x - 15 = 0 \)[/tex] are [tex]\( x = -5 \)[/tex], [tex]\( x = -\sqrt{3} \)[/tex], and [tex]\( x = \sqrt{3} \)[/tex].
To write this in the required form:
[tex]\[ x = -5, \quad \pm \sqrt{3} \][/tex]
So the values are:
- [tex]\( x = -5 \)[/tex]
- [tex]\( \pm \sqrt{3} \)[/tex]
Therefore, the solved equation yields [tex]\( x \)[/tex] values as [tex]\( -5 \)[/tex], [tex]\( -\sqrt{3} \)[/tex], and [tex]\( \sqrt{3} \)[/tex].
