https://file.aiquickdraw.com/mathbot/file/1725473783083e3tkx7st.jpg

from PIL import Image
import pytesseract

# Load the image
image_path = "/mnt/data/1725473783083e3tkx7st.jpg"
image = Image.open(image_path)

# Use OCR to extract text
text = pytesseract.image_to_string(image)
textIt seems that the text extraction from the image was not successful. The output is garbled and does not provide meaningful information.

I'll manually analyze the content of the image.

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**Problem Statement:**
Given a polynomial \( p(x) = x^7 - 3x^6 - 14x^5 - 14x^4 + 36x^3 + 43x^2 + 10x + 1 \), you need to determine which set contains only the possible rational zeros of the polynomial.

The **Rational Root Theorem** states that the possible rational roots of a polynomial are of the form:

\[
\text{Possible Rational Roots} = \pm \frac{\text{factors of constant term}}{\text{factors of leading coefficient}}
\]

For \( p(x) = x^7 - 3x^6 - 14x^5 - 14x^4 + 36x^3 + 43x^2 + 10x + 1 \):

- **Leading coefficient**: 1 (coefficient of \( x^7 \))
- **Constant term**: 1

Therefore, the possible rational roots are:

\[
\pm \frac{\text{factors of } 1}{\text{factors of } 1} = \pm 1
\]

So, the possible rational roots are \( \pm 1 \).

---

Let's evaluate the options:

- **Option A**: Contains \( \pm 1 \) among other incorrect values.
- **Option B**: Contains \( \pm 1 \) among other incorrect values.
- **Option C**: Contains \( \pm 1 \) among other incorrect values.
- **Option D**: Contains \( \pm 1 \) among other incorrect values.
- **Option E**: None of the options.

Since all the given sets contain other values not allowed by the Rational Root Theorem, **Option E** (None of the options) is correct.

Would you like any further details or have any other questions? Here are some additional related questions:

1. What is the Rational Root Theorem, and how is it applied?
2. How can we determine the factors of a polynomial like \( p(x) \)?
3. What are other methods to find the roots of a polynomial?
4. Why are the other sets in the options not valid?
5. Can irrational or complex roots also be found using similar techniques?

**Tip:** When using the Rational Root Theorem, always check both positive and negative factors of the constant term!

Which of the following sets contains only the possible rational zeros of p(x) = x^7 - 3x^6 - 14x^5 - 14x^4 + 36x^3 + 43x^2 + 10x + 1?

Explore how to determine the possible rational zeros of the polynomial p(x) = x^7 - 3x^6 - 14x^5 - 14x^4 + 36x^3 + 43x^2 + 10x + 1. This problem involves applying the Rational Root Theorem, a key concept in algebra, suitable for students in Grades 10-12.

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