To solve the given problem step by step, let's analyze the function and address each part of the question.

Function: $f(x) = (x + 3)(x + 1)^2(x - 1)$

(a) End behavior of $f(x)$:

The end behavior is determined by the degree of the polynomial and the leading coefficient.

• The degree is $1 + 2 + 1 = 4$ (sum of the exponents of the factors).
• The leading term is $x^4$ (since all terms contribute positive powers of $x$).
• Since the degree is even and the leading coefficient is positive, the end behavior is:
  • $f(x) \to +\infty$ as $x \to +\infty$
  • $f(x) \to +\infty$ as $x \to -\infty$

Answer: The end behavior of $f(x)$ is "up on both sides."

(b) Real zeros and their behavior:

1. Find the zeros:

   • From $(x + 3) = 0$, $x = -3$.
   • From $(x + 1)^2 = 0$, $x = -1$ (multiplicity 2).
   • From $(x - 1) = 0$, $x = 1$.

2. Behavior of the graph at each zero:

   • At $x = -3$: The factor $(x + 3)$ has multiplicity 1 (odd). The graph crosses the x-axis at this zero.
   • At $x = -1$: The factor $(x + 1)^2$ has multiplicity 2 (even). The graph touches the x-axis at this zero but does not cross.
   • At $x = 1$: The factor $(x - 1)$ has multiplicity 1 (odd). The graph crosses the x-axis at this zero.

Answer:

• Zeros where the graph crosses the x-axis: -3, 1.
• Zeros where the graph touches but does not cross the x-axis: -1.

(c) $y$-intercept:

The $y$-intercept is the value of $f(x)$ when $x = 0$.

$f(0) = (0 + 3)(0 + 1)^2(0 - 1) = (3)(1)^2(-1) = -3$

Answer: $y$-intercept is $-3$.

(d) Graph $f(x)$:

To graph $f(x) = (x + 3)(x + 1)^2(x - 1)$, follow these steps:

1. Plot the zeros:

   • Crosses at $x = -3$ and $x = 1$.
   • Touches at $x = -1$.

2. Mark the $y$-intercept: $(0, -3)$.

3. Sketch the end behavior:

   • Both ends of the graph point upwards.

4. Draw the graph:

   • The graph starts upward on the left.
   • Crosses the x-axis at $x = -3$, touches at $x = -1$, crosses again at $x = 1$.

Summary of Answers:

1. (a) End behavior: Up on both sides.
2. (b) Zeros:
   • Crosses the x-axis at: $-3, 1$.
   • Touches but does not cross at: $-1$.
3. (c) $y$-intercept: $-3$.

Would you like help with plotting this graph or verifying any part in detail?

5 Related Questions:

1. How do we determine multiplicity for a given factor?
2. What are the rules for identifying the end behavior of a polynomial?
3. Why does the graph touch but not cross the x-axis for even multiplicities?
4. How can we find turning points for this function?
5. Can we confirm these results by expanding the polynomial?

Tip: Always check the multiplicity of each zero to determine how the graph behaves at that point.