We can divide the polynomial \( 6x^3 - 5x^2 + 4x - 1 \) by \( 3x - 1 \) using long division. Here's the step-by-step process:

### Step 1: Divide the first term
Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{6x^3}{3x} = 2x^2
\]
Multiply \( 2x^2 \) by \( 3x - 1 \) to get:
\[
2x^2(3x - 1) = 6x^3 - 2x^2
\]
Subtract this from the original polynomial:
\[
(6x^3 - 5x^2 + 4x - 1) - (6x^3 - 2x^2) = -3x^2 + 4x - 1
\]

### Step 2: Divide the next term
Now, divide \( -3x^2 \) by \( 3x \):
\[
\frac{-3x^2}{3x} = -x
\]
Multiply \( -x \) by \( 3x - 1 \) to get:
\[
-x(3x - 1) = -3x^2 + x
\]
Subtract this from the current remainder:
\[
(-3x^2 + 4x - 1) - (-3x^2 + x) = 3x - 1
\]

### Step 3: Divide the last term
Now, divide \( 3x \) by \( 3x \):
\[
\frac{3x}{3x} = 1
\]
Multiply \( 1 \) by \( 3x - 1 \) to get:
\[
1(3x - 1) = 3x - 1
\]
Subtract this from the current remainder:
\[
(3x - 1) - (3x - 1) = 0
\]

### Final Answer:
The quotient is:
\[
2x^2 - x + 1
\]
There is no remainder.

Would you like further details or clarification?

Here are some related questions for further practice:
1. How would you divide \( 8x^3 - 2x^2 + 7x - 4 \) by \( 2x - 3 \)?
2. What is the process of dividing polynomials using synthetic division?
3. Can you apply the remainder theorem to this division problem?
4. How do you factor polynomials before dividing them?
5. How do polynomial long division and synthetic division differ?

**Tip:** Always align terms by degree when performing polynomial long division to avoid confusion.

Learn how to divide the polynomial 6x³-5x²+4x-1 by 3x-1 using the long division method. This detailed guide covers polynomial division step-by-step, from calculating the quotient to finding the remainder. Suitable for students in Grades 10-12.

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