If `f(x)` is divided by `(2x+3)`, then the remainder is :

Options

A. `f(2//3)`
B. `f(-2//3)`
C. `f(3//2)`
D. `f(-3//2)`

Answer

To find the remainder when the polynomial \( f(x) \) is divided by \( 2x + 3 \), we can use the Remainder Theorem. Here’s how to solve the problem step by step:

### Step 1: Identify the divisor
The divisor in this case is \( 2x + 3 \).

### Step 2: Set the divisor equal to zero
To find the value of \( x \) that will help us determine the remainder, we set the divisor equal to zero:
\[ 2x + 3 = 0 \]

### Step 3: Solve for \( x \)
Now, we solve for \( x \):
\[ 2x = -3 \\ x = -\frac{3}{2} \]

### Step 4: Substitute \( x \) into \( f(x) \)
According to the Remainder Theorem, the remainder when \( f(x) \) is divided by \( 2x + 3 \) is equal to \( f\left(-\frac{3}{2}\right) \).

### Step 5: Conclusion
Thus, the remainder is \( f\left(-\frac{3}{2}\right) \).

### Final Answer
The remainder when \( f(x) \) is divided by \( 2x + 3 \) is \( f\left(-\frac{3}{2}\right) \).

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