What number should be subtracted from `x^3 + 3x^2 - 8x + 14 ` so that on dividing it by x -2, the remainder is 10?

Answer

To solve the problem, we need to determine what number should be subtracted from the polynomial \( P(x) = x^3 + 3x^2 - 8x + 14 \) so that when it is divided by \( x - 2 \), the remainder is 10.

### Step-by-Step Solution:

1. **Identify the Polynomial and the Desired Remainder**:
We have the polynomial \( P(x) = x^3 + 3x^2 - 8x + 14 \) and we want the remainder when divided by \( x - 2 \) to be 10.

2. **Use the Remainder Theorem**:
According to the Remainder Theorem, the remainder of \( P(x) \) when divided by \( x - c \) is \( P(c) \). Here, \( c = 2 \).
\[ P(2) = 2^3 + 3(2^2) - 8(2) + 14 \]

3. **Calculate \( P(2) \)**:
\[ P(2) = 8 + 3(4) - 16 + 14 \]
\[ = 8 + 12 - 16 + 14 \]
\[ = 18 \]

4. **Determine the Required Subtraction**:
We want the remainder to be 10. Currently, \( P(2) = 18 \). To find the number that needs to be subtracted, we set up the equation:
\[ 18 - x = 10 \]
where \( x \) is the number to be subtracted.

5. **Solve for \( x \)**:
\[ x = 18 - 10 = 8 \]

6. **Conclusion**:
Therefore, the number that should be subtracted from the polynomial \( P(x) \) is **8**.

### Final Answer:
The number to be subtracted is **8**.

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