(6x³-5x²+4x-1)÷(3x-1) solve it in a long division the traditional way

Let's perform the long division of the polynomial \((6x^3 - 5x^2 + 4x - 1)\) by \((3x - 1)\) step by step.

### Step 1: Setup the Division
We want to divide:
\[
\frac{6x^3 - 5x^2 + 4x - 1}{3x - 1}
\]
The dividend is \(6x^3 - 5x^2 + 4x - 1\), and the divisor is \(3x - 1\).

### Step 2: Divide the First Terms
- Divide the first term of the dividend (\(6x^3\)) by the first term of the divisor (\(3x\)):
\[
\frac{6x^3}{3x} = 2x^2
\]
So, the first term in the quotient is \(2x^2\).

### Step 3: Multiply and Subtract
- Multiply the entire divisor \((3x - 1)\) by \(2x^2\):
\[
(3x - 1)(2x^2) = 6x^3 - 2x^2
\]
- Subtract this from the original polynomial:
\[
(6x^3 - 5x^2 + 4x - 1) - (6x^3 - 2x^2) = (-5x^2 - (-2x^2)) + 4x - 1 = -3x^2 + 4x - 1
\]

### Step 4: Divide the First Terms Again
- Now divide the first term of the new expression \(-3x^2\) by the first term of the divisor \(3x\):
\[
\frac{-3x^2}{3x} = -x
\]
So, the second term in the quotient is \(-x\).

### Step 5: Multiply and Subtract Again
- Multiply the entire divisor \((3x - 1)\) by \(-x\):
\[
(3x - 1)(-x) = -3x^2 + x
\]
- Subtract this from the current expression:
\[
(-3x^2 + 4x - 1) - (-3x^2 + x) = (4x - x) - 1 = 3x - 1
\]

### Step 6: Divide the First Terms Again
- Now divide the first term of the new expression \(3x\) by the first term of the divisor \(3x\):
\[
\frac{3x}{3x} = 1
\]
So, the third term in the quotient is \(1\).

### Step 7: Multiply and Subtract Again
- Multiply the entire divisor \((3x - 1)\) by \(1\):
\[
(3x - 1)(1) = 3x - 1
\]
- Subtract this from the current expression:
\[
(3x - 1) - (3x - 1) = 0
\]

### Final Answer:
Thus, the quotient is:
\[
2x^2 - x + 1
\]
and there is no remainder.

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

Would you like to see more details on any of these steps or have further questions?

Here are some related problems for practice:
1. Divide \(4x^3 + 2x^2 - 3x + 5\) by \(x + 2\).
2. Divide \(8x^3 - 6x^2 + 7x - 9\) by \(2x - 1\).
3. Perform long division on \((10x^3 - 15x^2 + 5x - 3)\) by \(5x - 2\).
4. Divide \(x^4 + x^3 - 2x + 1\) by \(x - 1\).
5. Perform synthetic division for \((x^3 - 5x + 6)\) divided by \((x - 2)\).

**Tip:** Always check your subtraction carefully during polynomial long division, as a small mistake can affect the entire solution.

Learn how to solve the polynomial division problem (6x³ - 5x² + 4x - 1) ÷ (3x - 1) using the traditional long division method. Follow the step-by-step solution to understand key algebraic concepts, including the division algorithm for polynomials. Suitable for students in Grades 10-12.

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