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Let's break down the given function \( y = (x - 2)(x + 3)^2(x + 1)^3 \) to find the zeros and the end behavior.

### 1. **Zeros of the function:**
Zeros of a function occur where \( y = 0 \), i.e., where any factor of the function equals zero.

- From \( (x - 2) \), the zero is \( x = 2 \).
- From \( (x + 3)^2 \), the zero is \( x = -3 \), with multiplicity 2 (since it's squared).
- From \( (x + 1)^3 \), the zero is \( x = -1 \), with multiplicity 3 (since it's cubed).

Thus, the **zeros** of the function are:
\[ x = 2, -3, -1 \]

The multiplicities are important in determining how the graph behaves near each zero:
- \( x = 2 \) has multiplicity 1, so the graph crosses the x-axis here.
- \( x = -3 \) has multiplicity 2, so the graph touches the x-axis and turns around (the zero is of even multiplicity).
- \( x = -1 \) has multiplicity 3, so the graph crosses the x-axis but flattens at this point (the zero is of odd multiplicity greater than 1).

### 2. **End behavior:**
To determine the end behavior, we analyze the degree of the polynomial and its leading term.

- The degree of the polynomial is the sum of the powers of all the factors:
  \[
  1 \ (\text{from } (x - 2)) + 2 \ (\text{from } (x + 3)^2) + 3 \ (\text{from } (x + 1)^3) = 6
  \]
  So, the polynomial is degree 6, which is an even degree.
- The leading term of the polynomial (ignoring constants) will be \( x^6 \), since each factor contributes to the highest power of \( x \).
- Since the leading term is \( x^6 \), which is positive and even, the **end behavior** of the function is:
  - As \( x \to +\infty \), \( y \to +\infty \)
  - As \( x \to -\infty \), \( y \to +\infty \)

Thus, the end behavior is:
- As \( x \to \infty \), \( y \to \infty \)
- As \( x \to -\infty \), \( y \to \infty \)

### Final Answer:
- The zeros are \( x = 2, -3, -1 \).
- The end behavior is: as \( x \to \infty \), \( y \to \infty \); as \( x \to -\infty \), \( y \to \infty \).

Would you like more details or have any questions on this? Here are some related questions for further exploration:

1. How does the multiplicity of a zero affect the graph at that point?
2. What would the graph look like near \( x = -1 \) and \( x = -3 \)?
3. How do we factor polynomials of higher degree when given in standard form?
4. How does the leading coefficient affect the end behavior of a polynomial?
5. Can you explain the difference in end behavior between even-degree and odd-degree polynomials?

**Tip:** Always check the multiplicity of zeros to understand how the graph interacts with the x-axis at each root.

Find the zeros and end behavior for the function y = (x - 2)(x + 3)^2(x + 1)^3.

Solve the polynomial function y = (x - 2)(x + 3)^2(x + 1)^3 by finding its zeros and determining its end behavior. Learn how multiplicity affects the graph's shape and how the degree of the polynomial influences the end behavior. Suitable for students in Grades 10-12.

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