What must be sutracted from `(x ^(3) + 4x ^(2) - 6x - 14)` to obtain a polynomial exactly divisible by `(x ^(2) - x - 2)` ?

Options

A. `x +8`
B. `x - 4`
C. `x +4`
D. `2x +4`

Answer

To solve the problem of finding what must be subtracted from the polynomial \( P(x) = x^3 + 4x^2 - 6x - 14 \) to make it exactly divisible by \( D(x) = x^2 - x - 2 \), we can follow these steps:

### Step 1: Perform Polynomial Long Division
We need to divide \( P(x) \) by \( D(x) \) to find the remainder.

1. **Divide the leading term** of \( P(x) \) by the leading term of \( D(x) \):
\[ \frac{x^3}{x^2} = x \]

2. **Multiply \( D(x) \)** by \( x \):
\[ x(x^2 - x - 2) = x^3 - x^2 - 2x \]

3. **Subtract this from \( P(x) \)**:
\[ (x^3 + 4x^2 - 6x - 14) - (x^3 - x^2 - 2x) = 4x^2 + x^2 - 6x + 2x - 14 \]
Simplifying gives:
\[ 5x^2 - 4x - 14 \]

### Step 2: Repeat the Division Process
Now we need to divide the new polynomial \( 5x^2 - 4x - 14 \) by \( D(x) \).

1. **Divide the leading term**:
\[ \frac{5x^2}{x^2} = 5 \]

2. **Multiply \( D(x) \)** by \( 5 \):
\[ 5(x^2 - x - 2) = 5x^2 - 5x - 10 \]

3. **Subtract this from \( 5x^2 - 4x - 14 \)**:
\[ (5x^2 - 4x - 14) - (5x^2 - 5x - 10) = -4x + 5x - 14 + 10 \]
Simplifying gives:
\[ x - 4 \]

### Step 3: Find What to Subtract
The remainder from the division is \( x - 4 \). To make \( P(x) \) exactly divisible by \( D(x) \), we need to subtract this remainder from \( P(x) \).

Thus, we conclude that:
\[ \text{What must be subtracted} = x - 4 \]

### Final Answer
To make the polynomial \( P(x) \) exactly divisible by \( D(x) \), we must subtract \( x - 4 \).

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