https://file.aiquickdraw.com/mathbot/file/1726972000314emll7fg0.jpg

Let's go through and solve each of the problems you requested from the image. 

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### Problem 13:  
**Polynomial: \( 7 - 2x^2 \)**

- **Degree**: The highest power of \( x \) is 2, so the degree is **2**.
- **Leading coefficient**: The coefficient of the highest degree term \( -2x^2 \) is **-2**.

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### Problem 15:  
**Polynomial: \( x(4 - x^2)(2x + 1) \)**

First, expand the expression:

1. Expand \( 4 - x^2 = (2 + x)(2 - x) \) (difference of squares)
2. Multiply \( x \cdot (2 - x)(2 + x) \cdot (2x + 1) \).

After expanding, the highest degree term will be \( -2x^4 \), so:
- **Degree**: 4 (the highest power of \( x \)).
- **Leading coefficient**: **-2**.

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### Problem 17:  
**Function: \( f(x) = x^4 \)**

- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).

**End behavior**:
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).

This is typical for even-degree polynomials with a positive leading coefficient.

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### Problem 21:  
**Function: \( f(x) = -2x^4 - 3x^2 + x - 1 \)**

- The leading term is \( -2x^4 \), so:
  - As \( x \to \infty \), \( f(x) \to -\infty \).
  - As \( x \to -\infty \), \( f(x) \to -\infty \).

**End behavior**:
- As \( x \to \infty \), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to -\infty \).

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### Problem 25:  
**Function: \( f(t) = 2(t - 1)(t + 2)(t - 3) \)**

To find the intercepts, set \( f(t) = 0 \):

- \( t - 1 = 0 \Rightarrow t = 1 \)
- \( t + 2 = 0 \Rightarrow t = -2 \)
- \( t - 3 = 0 \Rightarrow t = 3 \)

The intercepts are \( t = 1, -2, 3 \).

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### Problem 26:  
**Function: \( g(n) = -2(3n - 1)(2n + 1) \)**

Set \( g(n) = 0 \):

- \( 3n - 1 = 0 \Rightarrow n = \frac{1}{3} \)
- \( 2n + 1 = 0 \Rightarrow n = -\frac{1}{2} \)

The intercepts are \( n = \frac{1}{3}, -\frac{1}{2} \).

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### Problem 27:  
**Function: \( f(x) = x^4 - 16 \)**

Factor the polynomial:

\( f(x) = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4) \)

The real intercepts occur at \( x = 2 \) and \( x = -2 \).

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### Problem 28:  
**Function: \( f(x) = x^3 + 7 \)**

Set \( f(x) = 0 \):

\( x^3 + 7 = 0 \Rightarrow x^3 = -7 \Rightarrow x = \sqrt[3]{-7} \)

This gives a real intercept of \( x = \sqrt[3]{-7} \).

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### Problem 29:  
**Function: \( f(x) = x(x^2 - 2x - 8) \)**

Factor the quadratic:

\( f(x) = x(x - 4)(x + 2) \)

The intercepts are \( x = 0, 4, -2 \).

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### Problem 31 (Graphical):
The graph shows 3 distinct real roots and two turning points, suggesting at least degree **3**. Hence, the least possible degree is **3**.

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### Problem 33 (Graphical):
The graph shows 4 distinct real roots and three turning points, suggesting at least degree **4**. Hence, the least possible degree is **4**.

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Let me know if you'd like further details or clarifications on any of these problems!

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Here are some relative questions for practice:

1. What are the intercepts of \( f(x) = (x + 4)(x - 3)(x - 2) \)?
2. Determine the end behavior of \( f(x) = 3x^6 - 2x^4 + 7 \).
3. Factor \( f(x) = x^3 + 6x^2 + 9x \).
4. What is the degree of \( f(x) = (x + 1)^2(x - 2)^3 \)?
5. Determine the least degree of a polynomial graph with 3 turning points.

**Tip**: Always check the highest degree term to understand the polynomial’s end behavior and leading coefficient.

Solve polynomial problems by finding degrees, leading coefficients, end behavior, intercepts, and graphical polynomial analysis. This includes step-by-step solutions for problems 13, 15, 17, 21, 25, and more. Suitable for high school students in Grades 9-12.

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