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.

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!