Find the remainder when the polynomial `(5x^(3)-8x^(2)-2x-5)` is divided by x -2.

Options

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

Answer

To find the remainder when the polynomial \(5x^3 - 8x^2 - 2x - 5\) is divided by \(x - 2\), we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \(p(x)\) by \(x - a\) is \(p(a)\).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
- The polynomial is \(p(x) = 5x^3 - 8x^2 - 2x - 5\).
- The divisor is \(x - 2\), which means \(a = 2\).

2. **Evaluate the polynomial at \(x = 2\)**:
- We need to calculate \(p(2)\):
\[ p(2) = 5(2)^3 - 8(2)^2 - 2(2) - 5 \]

3. **Calculate \(2^3\) and \(2^2\)**:
- \(2^3 = 8\)
- \(2^2 = 4\)

4. **Substitute these values into the polynomial**:
\[ p(2) = 5(8) - 8(4) - 2(2) - 5 \]

5. **Perform the multiplications**:
\[ p(2) = 40 - 32 - 4 - 5 \]

6. **Combine the results**:
- First, calculate \(40 - 32 = 8\).
- Then, calculate \(8 - 4 = 4\).
- Finally, calculate \(4 - 5 = -1\).

7. **Conclusion**:
- The remainder when \(5x^3 - 8x^2 - 2x - 5\) is divided by \(x - 2\) is \(-1\).

### Final Answer:
The remainder is \(-1\).

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