One of the factors of `3x^(3)+x^(2)-12x-4` is :

Options

A. `x-2`
B. `3x-1`
C. `2x+1`
D. `3x+2`

Answer

To determine one of the factors of the polynomial \(3x^3 + x^2 - 12x - 4\), we can use the factor theorem, which states that if \(f(a) = 0\) for some polynomial \(f(x)\), then \(x - a\) is a factor of \(f(x)\).

### Step-by-Step Solution:

1. **Identify the polynomial**: We have the polynomial \(f(x) = 3x^3 + x^2 - 12x - 4\).

2. **List the options**: The options given are:
- (i) \(x - 2\)
- (ii) \(3x - 1\)
- (iii) \(2x + 1\)
- (iv) \(3x + 2\)

3. **Test the first option \(x - 2\)**:
- Set \(x - 2 = 0\) to find \(x = 2\).
- Substitute \(x = 2\) into the polynomial:
\[ f(2) = 3(2)^3 + (2)^2 - 12(2) - 4 \]
- Calculate each term:
\[ = 3(8) + 4 - 24 - 4 \]
\[ = 24 + 4 - 24 - 4 \]
\[ = 0 \]

4. **Conclusion**: Since \(f(2) = 0\), by the factor theorem, \(x - 2\) is a factor of the polynomial \(3x^3 + x^2 - 12x - 4\).

5. **Final Answer**: Therefore, one of the factors of \(3x^3 + x^2 - 12x - 4\) is \(x - 2\).