To analyze the function \( f(x) = x^3 + 13x^2 + 34x - 48 \), let's go through some key steps:

### 1. **Determine the Roots (Zeros) of the Function**
   To find the roots, we set \( f(x) = 0 \):
   \[
   x^3 + 13x^2 + 34x - 48 = 0
   \]
   Solving a cubic equation can be challenging, so we could start by using the Rational Root Theorem, synthetic division, or a numerical approach to find possible rational roots.

### 2. **Factorization**
   Once we find one root (say \( x = r \)), we can factor the cubic polynomial as \( (x - r) \times Q(x) \), where \( Q(x) \) is a quadratic equation. Solving \( Q(x) = 0 \) would then give the remaining roots.

### 3. **Determine the Derivative to Find Critical Points**
   To find the critical points (local maxima and minima), we calculate the first derivative:
   \[
   f'(x) = 3x^2 + 26x + 34
   \]
   Setting \( f'(x) = 0 \) gives the critical points.

### 4. **Analyze the Behavior of the Function**
   - **Intervals of Increase and Decrease**: By testing intervals around the critical points, we can determine where \( f(x) \) is increasing or decreasing.
   - **Concavity and Points of Inflection**: Taking the second derivative \( f''(x) = 6x + 26 \) and setting it to zero allows us to find points of inflection, where the concavity of \( f(x) \) changes.

Would you like a step-by-step solution to find the roots or further assistance with specific details?

### Further Exploration Questions:
1. How can the Rational Root Theorem help in solving \( f(x) = 0 \) for this cubic polynomial?
2. What are the practical steps for using synthetic division on this equation?
3. How does the derivative \( f'(x) = 3x^2 + 26x + 34 \) help determine the local extrema?
4. Why is it useful to find the points of inflection by solving \( f''(x) = 6x + 26 \)?
5. How can graphing \( f(x) \) support our understanding of its overall behavior?

**Tip:** For cubic equations, identifying one root simplifies the factorization process and leads to easier solutions for the remaining terms.

Explore the cubic function f(x) = x^3 + 13x^2 + 34x - 48. Learn how to find its roots, critical points, and intervals of increase or decrease through factorization, the Rational Root Theorem, and derivatives. This guide provides detailed insights into polynomial analysis for high school math students.

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