To analyze the function $y = x^3 + 3x^2 - 4x - 12$, let's determine the x- and y-intercepts, factor the polynomial if necessary, and identify any excluded regions.

1. Finding the x-intercepts:

The x-intercepts occur when $y = 0$, meaning we need to solve the equation: $x^3 + 3x^2 - 4x - 12 = 0$

Step 1: Factoring the polynomial

We will attempt factoring by grouping or using synthetic division. We will check for rational roots using the Rational Root Theorem.

The possible rational roots are the factors of the constant term (-12) divided by the factors of the leading coefficient (1): $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

Testing $x = -2$ using synthetic division:

$\text{Dividing } (x^3 + 3x^2 - 4x - 12) \text{ by } (x + 2):$

-2 & 1 & 3 & -4 & -12 \\ & & -2 & -2 & 12 \\ \hline & 1 & 1 & -6 & 0 \\ \end{array}$$ The result is $$x^2 + x - 6$$, and the remainder is zero, indicating that $$x = -2$$ is indeed a root, and we can factor the cubic polynomial as: $$(x + 2)(x^2 + x - 6)$$ #### Step 2: Factoring $$x^2 + x - 6$$ Next, factor the quadratic expression: $$x^2 + x - 6 = (x - 2)(x + 3)$$ So, the complete factorization of the cubic polynomial is: $$x^3 + 3x^2 - 4x - 12 = (x + 2)(x - 2)(x + 3)$$ #### Step 3: Solving for the x-intercepts Set each factor equal to zero to find the x-intercepts: $$x + 2 = 0 \quad \Rightarrow \quad x = -2$$ $$x - 2 = 0 \quad \Rightarrow \quad x = 2$$ $$x + 3 = 0 \quad \Rightarrow \quad x = -3$$ Thus, the x-intercepts are $$x = -2, 2, -3$$. ### 2. **Finding the y-intercept:** The y-intercept occurs when $$x = 0$$. Substituting $$x = 0$$ into the original equation: $$y = (0)^3 + 3(0)^2 - 4(0) - 12 = -12$$ Thus, the y-intercept is $$(0, -12)$$. ### 3. **Excluded regions:** Since this is a polynomial function, there are no vertical asymptotes or excluded regions. The domain of the function is all real numbers, $$(-\infty, \infty)$$. ### 4. **Graph sketch:** To sketch the graph, consider the following key points: - **x-intercepts** at $$(-3, 0), (-2, 0), (2, 0)$$ - **y-intercept** at $$(0, -12)$$ - Since the leading term is $$x^3$$, the end behavior is as follows: - As $$x \to -\infty$$, $$y \to -\infty$$ - As $$x \to \infty$$, $$y \to \infty$$ The curve passes through the intercepts and has a cubic shape, with a change in direction at each intercept. Would you like a visual sketch for this graph, or do you want further details on any step? --- ### 5 Related Questions: 1. How do you use synthetic division to factor a cubic polynomial? 2. How can you determine the end behavior of polynomial functions? 3. What is the Rational Root Theorem, and how is it applied? 4. How does the degree of a polynomial affect the number of intercepts and turning points? 5. How can you use factoring to solve higher-degree polynomials? ### Tip: For polynomials, always check for common factors and use synthetic division when possible to simplify the equation.