https://file.aiquickdraw.com/mathbot/file/1729593198412drzjjqxt.jpg

Let's go through the questions from the image you've provided.

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### **Part I: Multiple Choice**
1. **What should n be if f(x) = x^n defines a polynomial function?**
   - The answer is **a. an integer**, because for a polynomial function, the exponent (n) must be an integer.

2. **Which of the following is an example of a polynomial function?**
   - The correct answer is **a. f(x) = 4/x^3 + 3x - 1**, because the term 4/x^3 has a negative exponent when rewritten as 4x^(-3), which is not valid for a polynomial.

3. **What is the leading coefficient of the polynomial function f(x) = 2x + x^3 + 4?**
   - The leading term is **x^3**, and its coefficient is **1**. So, the answer is **c. 3**.

4. **How should the polynomial function f(x) = 2x + 3x^3 + 3x^5 + 4 be written in standard form?**
   - In standard form, terms are arranged in descending order of degree. So, the correct form is **f(x) = 3x^5 + 3x^3 + 2x + 4**, which corresponds to option **b**.

5. **What are the end behaviors of the graph of f(x) = 2x + x^3 + 3x^5 - 4?**
   - Since the degree of the leading term is 5 (an odd degree) and its coefficient is positive, the graph **falls to the left and rises to the right**, which is option **b**.

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### **Part II: Table Completion for the Function y = -x(x + 6)(3x - 4)**
The table requires you to find the values of \(y\) for given \(x\) values using the function \(y = -x(x + 6)(3x - 4)\).

Let's compute the values:

1. For \(x = -7\):
   \[
   y = -(-7)((-7) + 6)(3(-7) - 4) = -(7)(-1)(-25) = -175
   \]

2. For \(x = -3\):
   \[
   y = -(-3)((-3) + 6)(3(-3) - 4) = -(3)(3)(-13) = 117
   \]

3. For \(x = 1\):
   \[
   y = -(1)((1) + 6)(3(1) - 4) = -(1)(7)(-1) = 7
   \]

4. For \(x = 2\):
   \[
   y = -(2)((2) + 6)(3(2) - 4) = -(2)(8)(2) = -32
   \]

Now the table is:

| X  | Y   |
|----|-----|
| -7 | -175|
| -3 | 117 |
|  1 | 7   |
|  2 | -32 |

The intercepts:
- **X-intercepts**: These occur where \(y = 0\). Solving \( -x(x + 6)(3x - 4) = 0 \), we get three solutions:
  - \(x = 0\), \(x = -6\), and \(x = \frac{4}{3}\).
  
- **Y-intercepts**: This occurs when \(x = 0\):
  \[
  y = -(0)((0) + 6)(3(0) - 4) = 0.
  \]
  So the Y-intercept is \((0, 0)\).

**Ordered pairs**: \((-7, -175), (-3, 117), (1, 7), (2, -32)\).

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### **Part III: Polynomial Function**

The function provided is \(y = x(x + 6)(3x - 4)\). We've already calculated the \(y\)-values for different \(x\)-values above, so this table should match the same results.

Let me know if you want further details on solving or have any specific questions! Here's a related expansion:

1. How would you find the degree of a polynomial function?
2. What are the steps to solve for intercepts algebraically?
3. Can you explain how the leading coefficient affects the graph's behavior?
4. How does the behavior of odd-degree polynomial functions differ from even-degree ones?
5. What would happen if a negative coefficient appeared before the highest-degree term?

**Tip**: To find end behavior, always focus on the highest-degree term and its coefficient in the polynomial!

Given the function y = -x(x + 6)(3x - 4), find the value of y for different x values and solve related questions.

Learn how to solve the polynomial y = -x(x + 6)(3x - 4) for different values of x. This problem involves finding y-values, determining x- and y-intercepts, and understanding end behavior for polynomials. Suitable for high school students studying algebra.

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