What is the remainder when `(4x ^(3) - 3x ^(2) + 2x -1)` is divided by (x +2) ?

Options

A. 48
B. 49
C. `-49`
D. `-48`

Answer

To find the remainder when \(4x^3 - 3x^2 + 2x - 1\) is divided by \(x + 2\), we can use the Remainder Theorem. According to the theorem, the remainder of a polynomial \(f(x)\) when divided by \(x - c\) is \(f(c)\). In our case, we will evaluate the polynomial at \(c = -2\) (since we are dividing by \(x + 2\)).

### Step-by-Step Solution:

1. **Identify the polynomial and the value of c**:
The polynomial is \(f(x) = 4x^3 - 3x^2 + 2x - 1\) and we need to evaluate it at \(c = -2\).

2. **Substitute \(c = -2\) into the polynomial**:
\[ f(-2) = 4(-2)^3 - 3(-2)^2 + 2(-2) - 1 \]

3. **Calculate each term**:
- Calculate \(4(-2)^3\):
\[ 4(-2)^3 = 4 \times (-8) = -32 \]
- Calculate \(-3(-2)^2\):
\[ -3(-2)^2 = -3 \times 4 = -12 \]
- Calculate \(2(-2)\):
\[ 2(-2) = -4 \]
- The constant term is \(-1\).

4. **Combine all the terms**:
\[ f(-2) = -32 - 12 - 4 - 1 \]

5. **Perform the addition**:
- First, combine \(-32\) and \(-12\):
\[ -32 - 12 = -44 \]
- Next, combine \(-44\) and \(-4\):
\[ -44 - 4 = -48 \]
- Finally, combine \(-48\) and \(-1\):
\[ -48 - 1 = -49 \]

6. **Conclusion**:
The remainder when \(4x^3 - 3x^2 + 2x - 1\) is divided by \(x + 2\) is \(-49\).

### Final Answer:
The remainder is \(-49\).