For a polynomial, dividend is `x^(4)+4x-2x^(2)+x^(3)-10`, quotient is `x^(3)+3x^(2)+4x+12` and remainder is 14, then divisor is equal to

Options

A. `x^(2)+2`
B. `x^(2)-2`
C. `x+2`
D. `x-2`

Answer

To find the divisor given the dividend, quotient, and remainder, we can use the polynomial division formula:

**Dividend = Divisor × Quotient + Remainder**

Given:
- Dividend: \( x^4 + 4x - 2x^2 + x^3 - 10 \)
- Quotient: \( x^3 + 3x^2 + 4x + 12 \)
- Remainder: 14

We need to find the divisor. Rearranging the formula gives us:

**Divisor = (Dividend - Remainder) / Quotient**

### Step-by-Step Solution:

1. **Write down the expression for the dividend and remainder**:
\[ \text{Dividend} = x^4 + 4x - 2x^2 + x^3 - 10 \]
\[ \text{Remainder} = 14 \]

2. **Substitute the values into the formula**:
\[ \text{Divisor} = \frac{\text{Dividend} - \text{Remainder}}{\text{Quotient}} \]
\[ \text{Divisor} = \frac{(x^4 + 4x - 2x^2 + x^3 - 10) - 14}{x^3 + 3x^2 + 4x + 12} \]

3. **Simplify the numerator**:
\[ \text{Dividend} - \text{Remainder} = x^4 + x^3 - 2x^2 + 4x - 10 - 14 \]
\[ = x^4 + x^3 - 2x^2 + 4x - 24 \]

4. **Now substitute back into the divisor formula**:
\[ \text{Divisor} = \frac{x^4 + x^3 - 2x^2 + 4x - 24}{x^3 + 3x^2 + 4x + 12} \]

5. **Perform polynomial long division**:
- Divide \( x^4 \) by \( x^3 \) to get \( x \).
- Multiply \( x \) by the entire quotient \( (x^3 + 3x^2 + 4x + 12) \):
\[ x(x^3 + 3x^2 + 4x + 12) = x^4 + 3x^3 + 4x^2 + 12x \]
- Subtract this from the numerator:
\[ (x^4 + x^3 - 2x^2 + 4x - 24) - (x^4 + 3x^3 + 4x^2 + 12x) = -2x^3 - 6x^2 - 8x - 24 \]

6. **Now divide \(-2x^3\) by \(x^3\) to get \(-2\)**:
- Multiply \(-2\) by the entire quotient:
\[ -2(x^3 + 3x^2 + 4x + 12) = -2x^3 - 6x^2 - 8x - 24 \]
- Subtract this from the previous result:
\[ (-2x^3 - 6x^2 - 8x - 24) - (-2x^3 - 6x^2 - 8x - 24) = 0 \]

7. **Conclusion**:
Since the remainder is 0, we conclude that the divisor is:
\[ x + 2 \]

### Final Answer:
The divisor is \( x + 2 \).