https://file.aiquickdraw.com/mathbot/file/1726659577924t0984sob.jpg

The text extracted from the image wasn't fully clear. However, based on the visual content, the problem appears to be:

\[
\text{Divide } (6x^3 - 5x^2 + 4x - 1) \text{ by } (3x - 1)
\]

Let's solve this polynomial division using the long division method.

### Steps for Long Division:
1. **Divide the first term** of the numerator by the first term of the divisor:
   \[
   \frac{6x^3}{3x} = 2x^2
   \]
2. Multiply \( 2x^2 \) by the entire divisor \( (3x - 1) \):
   \[
   2x^2(3x - 1) = 6x^3 - 2x^2
   \]
3. Subtract this result from the original polynomial:
   \[
   (6x^3 - 5x^2 + 4x - 1) - (6x^3 - 2x^2) = (-5x^2 + 2x^2) + 4x - 1 = -3x^2 + 4x - 1
   \]
4. **Repeat the process** with the new polynomial \( -3x^2 + 4x - 1 \).

Let me continue the full calculation.The result of the division of \( (6x^3 - 5x^2 + 4x - 1) \) by \( (3x - 1) \) using the long division method is:

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

The remainder is \( 0 \), meaning the division is exact.

Would you like further details on any step or have any questions? Here are some related questions for practice:

1. How do you perform polynomial long division for higher degrees?
2. What happens if the remainder is not zero?
3. Can synthetic division be used here, and how does it differ?
4. What if the divisor was a binomial of higher degree?
5. How do you check the result of a polynomial division?

**Tip:** Always verify your result by multiplying the quotient by the divisor and checking if you get the original dividend.

Divide (6x^3 - 5x^2 + 4x - 1) by (3x - 1) using the long division method.

Solve the polynomial division problem where (6x^3 - 5x^2 + 4x - 1) is divided by (3x - 1) using the long division method. This example includes step-by-step calculations, making it ideal for students in Grades 9-11.

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