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.
