`p(x)=12x^(4)+13x^(3)-35x^(2)-16x+20`
Which of the following is a factor of the polynomial given above?

Options

A. `2x+3`
B. `3x-2`
C. `4x+5`
D. `5x-3`

Explanation

To determine which of the given options is a factor of the polynomial \( p(x) = 12x^4 + 13x^3 - 35x^2 - 16x + 20 \), we can use the Rational Root Theorem and perform polynomial long division.

### Step-by-Step Solution:

1. **Identify the Constant Term**:
The constant term of the polynomial \( p(x) \) is \( 20 \).

2. **List the Options**:
Let's assume the options provided are:
- Option A: \( 3x + 3 \)
- Option B: \( 3x - 2 \)
- Option C: \( 5x + 4 \)
- Option D: \( 3x - 3 \)

3. **Check Divisibility of Constant Terms**:
We will check if the constant term \( 20 \) is divisible by the constant terms of each option:
- For Option A: \( 3 \) (not divisible)
- For Option B: \( -2 \) (divisible, \( 20 / -2 = -10 \))
- For Option C: \( 4 \) (divisible, \( 20 / 4 = 5 \))
- For Option D: \( -3 \) (not divisible)

After this step, we can eliminate Options A and D since \( 3 \) and \( -3 \) do not divide \( 20 \).

4. **Remaining Options**:
The remaining options are B: \( 3x - 2 \) and C: \( 5x + 4 \).

5. **Perform Long Division for Option B**:
We will perform polynomial long division of \( p(x) \) by \( 3x - 2 \).

- Divide \( 12x^4 \) by \( 3x \) to get \( 4x^3 \).
- Multiply \( 4x^3 \) by \( (3x - 2) \) to get \( 12x^4 - 8x^3 \).
- Subtract this from \( p(x) \):
\[ (12x^4 + 13x^3) - (12x^4 - 8x^3) = 21x^3 \]
- Bring down the next term, \( -35x^2 \), to get \( 21x^3 - 35x^2 \).
- Divide \( 21x^3 \) by \( 3x \) to get \( 7x^2 \).
- Multiply \( 7x^2 \) by \( (3x - 2) \) to get \( 21x^3 - 14x^2 \).
- Subtract:
\[ (-35x^2 + 14x^2) = -21x^2 \]
- Bring down the next term, \( -16x \), to get \( -21x^2 - 16x \).
- Divide \( -21x^2 \) by \( 3x \) to get \( -7x \).
- Multiply \( -7x \) by \( (3x - 2) \) to get \( -21x^2 + 14x \).
- Subtract:
\[ (-16x - 14x) = -30x \]
- Bring down \( 20 \) to get \( -30x + 20 \).
- Divide \( -30x \) by \( 3x \) to get \( -10 \).
- Multiply \( -10 \) by \( (3x - 2) \) to get \( -30x + 20 \).
- Subtract:
\[ (20 - 20) = 0 \]

6. **Conclusion**:
Since the remainder is \( 0 \), \( 3x - 2 \) is indeed a factor of the polynomial \( p(x) \).

### Final Answer:
The factor of the polynomial \( p(x) \) is \( 3x - 2 \).

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