Answer:
f = 2.
Explanation:
We can use synthetic division to solve this problem.
First, we set up our synthetic division table:
1/2 | 2 5 -6 -2 f
------|----------------------------
| 1 3 -1
|----------------------------
2 6 -3 -3 f - 2
The numbers on the top row of the table are the coefficients of the terms in the polynomial, in descending order. For example, 2 is the coefficient of x^4, 5 is the coefficient of x^3, and so on.
The number on the left side of the table is the root we're dividing by, which is 1/2 in this case because 2x - 1 is a factor.
The first step in synthetic division is to "bring down" the first coefficient, which is 2 in this case.
Next, we multiply the root (1/2) by the first number we brought down (2), which gives us 1. We write this in the next row of the table.
We then add the 1 to the next coefficient, which is 5. This gives us 6, which we write in the next row. We repeat this process for the remaining coefficients, always adding the result to the next coefficient:
1/2 | 2 5 -6 -2 f
------|----------------------------
| 1 3 -1
|----------------------------
2 6 -3 -3 f - 2
The last number in the table, f - 2, is the remainder of the division. Since 2x - 1 is a factor, the remainder must be 0:
f - 2 = 0
Therefore, f = 2.