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.