https://file.aiquickdraw.com/mathbot/file/17295754280996d9flj1m.jpg

The function given in the image is:

\[
y = (x - 2)(x + 3)^2(x + 1)^3
\]

### 1. Finding the Zeros
To find the zeros, we set \( y = 0 \) and solve for \( x \). The zeros occur where any of the factors is equal to zero:

- \( (x - 2) = 0 \) ⟶ \( x = 2 \)
- \( (x + 3)^2 = 0 \) ⟶ \( x = -3 \) (multiplicity 2)
- \( (x + 1)^3 = 0 \) ⟶ \( x = -1 \) (multiplicity 3)

Thus, the zeros are:
- \( x = 2 \) (multiplicity 1),
- \( x = -3 \) (multiplicity 2),
- \( x = -1 \) (multiplicity 3).

### 2. End Behavior
To determine the end behavior of the polynomial function, we analyze the degree and the leading coefficient. The degree of the polynomial is the sum of the exponents of each factor:
- From \( (x - 2) \), the exponent is 1.
- From \( (x + 3)^2 \), the exponent is 2.
- From \( (x + 1)^3 \), the exponent is 3.

So, the total degree is \( 1 + 2 + 3 = 6 \).

Since the highest degree is even (6), and the leading coefficient (from expanding the product) is positive, the end behavior will be:
- As \( x \to \infty \), \( y \to \infty \).
- As \( x \to -\infty \), \( y \to \infty \).

This is typical of even-degree polynomials with positive leading coefficients, where both ends of the graph point upwards.

### Summary:
- **Zeros:** \( x = 2 \), \( x = -3 \), and \( x = -1 \).
- **End Behavior:** \( y \to \infty \) as \( x \to \infty \) and \( y \to \infty \) as \( x \to -\infty \).

Would you like more details on any of these steps?

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Here are 5 follow-up questions to expand on this topic:
1. How does the multiplicity of a zero affect the shape of the graph at that point?
2. Can you find the y-intercept of the function?
3. What is the behavior of the graph near the zero at \( x = -1 \)?
4. How does the end behavior change if the leading coefficient is negative?
5. What is the importance of the degree in determining the general shape of a polynomial graph?

**Tip:** The multiplicity of a zero determines whether the graph touches or crosses the x-axis at that point. Odd multiplicities result in crossing, while even multiplicities result in touching and turning back.

Find the zeroes and end behavior for the following function: y = (x - 2)(x + 3)^2(x + 1)^3

Learn how to find the zeros and determine the end behavior of the polynomial function y = (x - 2)(x + 3)^2(x + 1)^3. This detailed solution covers important algebraic concepts such as zero product property and the end behavior of polynomials based on degree and leading coefficient. Ideal for high school students in Grades 9-12.

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