https://file.aiquickdraw.com/mathbot/file/1725492549778zjc83orf.jpg

I see that you've uploaded an image with a set of rational expressions and polynomial fractions. Let me go ahead and extract the content and solve each equation for you.

The equations appear to be fractions with polynomials in the numerator and denominator. I'll analyze them and provide step-by-step solutions.

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Here are the expressions from the image:

1. \(\frac{3x^2 - 3x + 12}{3x}\)
2. \(\frac{11x^9 + 3x^7 - 2x}{x}\)
3. \(\frac{8x^3 - x^2 + 2x - 1}{2x}\)
4. \(\frac{x^7 + 5x + 6}{x^2}\)
5. \(\frac{x^8 - 4x^4 + 3x^2}{5x^2}\)
6. \(\frac{x^5 + 6x - 1}{3x}\)
7. \(\frac{3x^9 - 15x^7 + 6x}{9x}\)

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Now I'll solve them one by one:

### 1. \(\frac{3x^2 - 3x + 12}{3x}\)

We can simplify this by dividing each term in the numerator by \(3x\):

\[
\frac{3x^2}{3x} - \frac{3x}{3x} + \frac{12}{3x}
= x - 1 + \frac{4}{x}
\]

So the simplified form is:
\[
x - 1 + \frac{4}{x}
\]

### 2. \(\frac{11x^9 + 3x^7 - 2x}{x}\)

Again, divide each term in the numerator by \(x\):

\[
\frac{11x^9}{x} + \frac{3x^7}{x} - \frac{2x}{x}
= 11x^8 + 3x^6 - 2
\]

So the simplified form is:
\[
11x^8 + 3x^6 - 2
\]

### 3. \(\frac{8x^3 - x^2 + 2x - 1}{2x}\)

Divide each term in the numerator by \(2x\):

\[
\frac{8x^3}{2x} - \frac{x^2}{2x} + \frac{2x}{2x} - \frac{1}{2x}
= 4x^2 - \frac{1}{2}x + 1 - \frac{1}{2x}
\]

So the simplified form is:
\[
4x^2 - \frac{1}{2}x + 1 - \frac{1}{2x}
\]

### 4. \(\frac{x^7 + 5x + 6}{x^2}\)

Divide each term in the numerator by \(x^2\):

\[
\frac{x^7}{x^2} + \frac{5x}{x^2} + \frac{6}{x^2}
= x^5 + \frac{5}{x} + \frac{6}{x^2}
\]

So the simplified form is:
\[
x^5 + \frac{5}{x} + \frac{6}{x^2}
\]

### 5. \(\frac{x^8 - 4x^4 + 3x^2}{5x^2}\)

Divide each term in the numerator by \(5x^2\):

\[
\frac{x^8}{5x^2} - \frac{4x^4}{5x^2} + \frac{3x^2}{5x^2}
= \frac{x^6}{5} - \frac{4x^2}{5} + \frac{3}{5}
\]

So the simplified form is:
\[
\frac{x^6}{5} - \frac{4x^2}{5} + \frac{3}{5}
\]

### 6. \(\frac{x^5 + 6x - 1}{3x}\)

Divide each term in the numerator by \(3x\):

\[
\frac{x^5}{3x} + \frac{6x}{3x} - \frac{1}{3x}
= \frac{x^4}{3} + 2 - \frac{1}{3x}
\]

So the simplified form is:
\[
\frac{x^4}{3} + 2 - \frac{1}{3x}
\]

### 7. \(\frac{3x^9 - 15x^7 + 6x}{9x}\)

Divide each term in the numerator by \(9x\):

\[
\frac{3x^9}{9x} - \frac{15x^7}{9x} + \frac{6x}{9x}
= \frac{x^8}{3} - \frac{5x^6}{3} + \frac{2}{3}
\]

So the simplified form is:
\[
\frac{x^8}{3} - \frac{5x^6}{3} + \frac{2}{3}
\]

---

Let me know if you'd like further clarification on any of these steps, or if you have any additional questions!

### Related Questions:
1. How do you simplify rational expressions with higher degree polynomials?
2. What are the common mistakes made when simplifying fractions with variables?
3. How do you handle terms with negative exponents in a rational expression?
4. Can all rational expressions be simplified?
5. How do you factor polynomials to simplify complex fractions?

### Tip:
Always check if you can factor the numerator and denominator before simplifying to avoid missing any simplifications.

Simplify the following polynomial fractions: 1. (3x^2 - 3x + 12) / 3x, 2. (11x^9 + 3x^7 - 2x) / x, 3. (8x^3 - x^2 + 2x - 1) / 2x, 4. (x^7 + 5x + 6) / x^2, 5. (x^8 - 4x^4 + 3x^2) / 5x^2, 6. (x^5 + 6x - 1) / 3x, 7. (3x^9 - 15x^7 + 6x) / 9x.

Learn how to simplify polynomial fractions step by step, such as (3x^2 - 3x + 12) / 3x and (11x^9 + 3x^7 - 2x) / x. This tutorial covers rational expressions, polynomial division, and simplification techniques for students in Grades 8-10.

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