The problem in the image involves analyzing the polynomial function $f(x) = x^2(x - 2)^3(x + 3)$ and answering several questions regarding its graph and x-intercepts.

a) End behavior of the graph using the Leading Coefficient Test:

The given polynomial function is $f(x) = x^2(x - 2)^3(x + 3)$.

1. The degree of the polynomial can be determined by adding the exponents:

   • $x^2$ has a degree of 2,
   • $(x - 2)^3$ has a degree of 3,
   • $(x + 3)$ has a degree of 1.

   So the total degree of the polynomial is $2 + 3 + 1 = 6$, which is even.

2. The leading term, which dominates the end behavior, comes from multiplying the highest degree terms from each factor:

   • The leading term is $x^6$, which has a positive coefficient.

For even degree polynomials with a positive leading coefficient, the end behavior is that the graph rises to the left and rises to the right.

Thus, option C (The graph rises to the left and to the right) is the correct choice.

b) X-intercepts of the graph:

The x-intercepts are found by setting $f(x) = 0$. From the factored form $f(x) = x^2(x - 2)^3(x + 3)$, the x-intercepts occur when each factor equals zero:

• $x^2 = 0$ gives $x = 0$,
• $(x - 2)^3 = 0$ gives $x = 2$,
• $(x + 3) = 0$ gives $x = -3$.

Thus, the x-intercepts are $0$, $2$, and $-3$.

c) Where the graph crosses the x-axis:

To determine where the graph crosses the x-axis, we need to check the multiplicity of the roots:

• $x = 0$ has a multiplicity of 2 (from $x^2$),
• $x = 2$ has a multiplicity of 3 (from $(x - 2)^3$),
• $x = -3$ has a multiplicity of 1 (from $(x + 3)$).

A root with an odd multiplicity means the graph crosses the x-axis at that point, while a root with even multiplicity means the graph touches the x-axis and turns around.

• The graph touches the x-axis at $x = 0$ (even multiplicity),
• The graph crosses the x-axis at $x = 2$ and $x = -3$ (odd multiplicities).

Thus, the correct answer is:

• The graph crosses the x-axis at $x = -3$ and $x = 2$.

Let me know if you need further clarifications!

Relative Questions:

1. What does the degree of a polynomial tell you about its end behavior?
2. How do you determine the multiplicity of a root from the factored form of a polynomial?
3. Why does the graph only cross the x-axis at certain points?
4. How can the Leading Coefficient Test help predict the shape of the graph?
5. What are the implications of a root having even multiplicity on the behavior of the graph?

Tip:

When analyzing polynomial graphs, always check the degree and leading coefficient first to predict end behavior and investigate root multiplicities to understand how the graph interacts with the x-axis.