https://file.aiquickdraw.com/mathbot/file/17302473853087o1739uk.jpg

To determine whether \( x - 5 \) is a factor of the polynomial \( 3x^3 + 8x^2 - 39x - 20 \), we need to look at the results of the synthetic division provided in the image. 

The synthetic division layout shows:

\[
-5 \,|\, 3 \quad 8 \quad -39 \quad -20
\]
\[
\phantom{-5 \,|\, } \quad -15 \quad 35 \quad 20
\]
\[
\phantom{-5 \,|\, } 3 \quad -7 \quad -4 \quad 0
\]

The remainder, located at the far right of the bottom row, is **0**. When the remainder is zero, it confirms that the divisor \( x - 5 \) divides the polynomial evenly, indicating that \( x - 5 \) is indeed a factor of the polynomial.

Thus, the correct answer is:

**A. \( x - 5 \) is a factor of \( 3x^3 + 8x^2 - 39x - 20 \)**.

Would you like a breakdown of the synthetic division steps, or have any other questions?

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**Follow-up questions to explore:**

1. How does synthetic division work for other divisors, such as \( x + 3 \)?
2. What does it mean if the remainder is not zero in synthetic division?
3. How would we find the other factors of \( 3x^3 + 8x^2 - 39x - 20 \) using synthetic division?
4. Can we verify that \( x - 5 \) is a factor using the Factor Theorem?
5. How does synthetic division relate to polynomial long division?

**Tip:** Synthetic division is especially useful when checking if linear factors, like \( x - 5 \), divide a polynomial evenly because it saves time compared to polynomial long division.

Choose the correct conclusion based on the right side synthetic division result provided in the image.

Learn how to determine if x - 5 is a factor of the polynomial 3x^3 + 8x^2 - 39x - 20 using synthetic division. This problem illustrates the process and explains the significance of a zero remainder in synthetic division, using the Factor Theorem. Suitable for students in Grades 10-12.

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