When ` f(x) = 12 x^(3) - 13 x^(2) - 5 x + 7 ` is divided by `( 3 x + 2)` then the remainder is

Options

A. 2
B. 0
C. `-1`
D. 1

Answer

To find the remainder when \( f(x) = 12x^3 - 13x^2 - 5x + 7 \) is divided by \( 3x + 2 \), we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \( f(x) \) by a linear polynomial \( ax + b \) can be found by evaluating \( f\left(-\frac{b}{a}\right) \).

### Step-by-Step Solution:

1. **Identify the coefficients**: In our case, the linear polynomial is \( 3x + 2 \). Here, \( a = 3 \) and \( b = 2 \).

2. **Calculate \( -\frac{b}{a} \)**:
\[ -\frac{b}{a} = -\frac{2}{3} \]

3. **Evaluate \( f\left(-\frac{2}{3}\right) \)**:
\[ f\left(-\frac{2}{3}\right) = 12\left(-\frac{2}{3}\right)^3 - 13\left(-\frac{2}{3}\right)^2 - 5\left(-\frac{2}{3}\right) + 7 \]

4. **Calculate each term**:
- First term:
\[ 12\left(-\frac{2}{3}\right)^3 = 12 \cdot -\frac{8}{27} = -\frac{96}{27} \]
- Second term:
\[ -13\left(-\frac{2}{3}\right)^2 = -13 \cdot \frac{4}{9} = -\frac{52}{9} \]
- Third term:
\[ -5\left(-\frac{2}{3}\right) = \frac{10}{3} \]
- Fourth term:
\[ 7 = \frac{63}{9} \]

5. **Combine all terms**:
- Convert all fractions to have a common denominator (27):
- First term: \( -\frac{96}{27} \)
- Second term: \( -\frac{52}{9} = -\frac{156}{27} \)
- Third term: \( \frac{10}{3} = \frac{90}{27} \)
- Fourth term: \( 7 = \frac{189}{27} \)

Now, combine them:
\[ f\left(-\frac{2}{3}\right) = -\frac{96}{27} - \frac{156}{27} + \frac{90}{27} + \frac{189}{27} \]
\[ = \frac{-96 - 156 + 90 + 189}{27} = \frac{27}{27} = 1 \]

6. **Conclusion**: The remainder when \( f(x) \) is divided by \( 3x + 2 \) is \( 1 \).

### Final Answer:
The remainder is \( 1 \).