use polynomial long division to divide 6x^3-5x^2+13x-4 by 3x-1.

a.x^2-4x+3
b.2x^2-x+4
c.2x^2+7/3x+16/9
d.2x^2+2x+5

Let's perform polynomial long division to divide \( 6x^3 - 5x^2 + 13x - 4 \) by \( 3x - 1 \).

### Step 1: Divide the leading term of the dividend by the leading term of the divisor
Divide \( 6x^3 \) by \( 3x \) to get \( 2x^2 \).

### Step 2: Multiply the entire divisor by this result
Multiply \( 2x^2 \) by \( 3x - 1 \):
\[
2x^2 \times (3x - 1) = 6x^3 - 2x^2
\]

### Step 3: Subtract this from the original polynomial
Subtract \( 6x^3 - 2x^2 \) from \( 6x^3 - 5x^2 + 13x - 4 \):
\[
(6x^3 - 5x^2 + 13x - 4) - (6x^3 - 2x^2) = -3x^2 + 13x - 4
\]

### Step 4: Repeat the process with the new polynomial
Now divide \( -3x^2 \) by \( 3x \) to get \( -x \).

Multiply \( -x \) by \( 3x - 1 \):
\[
-x \times (3x - 1) = -3x^2 + x
\]

Subtract this from the new polynomial:
\[
(-3x^2 + 13x - 4) - (-3x^2 + x) = 12x - 4
\]

### Step 5: Divide and repeat
Divide \( 12x \) by \( 3x \) to get \( 4 \).

Multiply \( 4 \) by \( 3x - 1 \):
\[
4 \times (3x - 1) = 12x - 4
\]

Subtract this from the current polynomial:
\[
(12x - 4) - (12x - 4) = 0
\]

### Final Answer
The quotient is \( 2x^2 - x + 4 \).

So the correct answer is **b. \( 2x^2 - x + 4 \)**.

Would you like details or have any questions?

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**Related Questions:**
1. How do you check the result of polynomial long division?
2. What is the remainder when dividing \( x^3 + 2x^2 - 5x + 6 \) by \( x + 1 \)?
3. How can synthetic division be used for the same problem?
4. What is the difference between polynomial long division and synthetic division?
5. How do you find the roots of the polynomial \( 6x^3 - 5x^2 + 13x - 4 \)?

**Tip:** After performing polynomial long division, you can always multiply the quotient by the divisor and add any remainder to verify your answer.

Learn how to use polynomial long division to divide 6x^3 - 5x^2 + 13x - 4 by 3x - 1 with a detailed step-by-step solution. This problem involves advanced algebraic concepts and is suitable for advanced high school students.

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