Divide 4x3-3x+1 by 2x-1

To divide 4x³-3x+1 by 2x-1, we can use polynomial long division.

First, we set up the division like this:

         2x² + x - 1
    -----------------
2x - 1 | 4x³ + 0x² - 3x + 1

Next, we look at the leading term of the dividend (4x³) and the leading term of the divisor (2x) and ask: "How many times does 2x go into 4x³?" The answer is 2x², so we write that above the division line and multiply by the divisor:

         2x² + x - 1
    -----------------
2x - 1 | 4x³ + 0x² - 3x + 1
       - 4x³ + 2x²
       -------------
             2x² - 3x

We then subtract the result from the dividend and bring down the next term (1x):

         2x² + x - 1
    -----------------
2x - 1 | 4x³ + 0x² - 3x + 1
       - 4x³ + 2x²
       -------------
             2x² - 3x
             - 2x² + x
             ----------
                   -2x + 1

We repeat the process with the new polynomial (-2x+1) and the divisor (2x-1):

         2x² + x - 1
    -----------------
2x - 1 | 4x³ + 0x² - 3x + 1
       - 4x³ + 2x²
       -------------
             2x² - 3x
             - 2x² + x
             ----------
                   -2x + 1
                   -2x + 1
                   ------
                        0

We end up with a remainder of 0, which means that the division is exact. Therefore, the quotient is:

2x² + x - 1

So the answer is (B) 2x²-x-1.