If `3x^(3) -2x^(2)y -13xy^(2)+10y^(3)` is divided by `x-2y`, then what will be the remainder?

Options

A. 0
B. x
C. `y+5`
D. `x-3`

Answer

To find the remainder when the polynomial \(3x^3 - 2x^2y - 13xy^2 + 10y^3\) is divided by \(x - 2y\), we can use the Remainder Theorem. According to the theorem, the remainder of a polynomial \(P(x)\) when divided by \(x - a\) is \(P(a)\). In our case, we will substitute \(x = 2y\) into the polynomial.

### Step-by-Step Solution:

1. **Identify the polynomial**:
\[ P(x) = 3x^3 - 2x^2y - 13xy^2 + 10y^3 \]

2. **Substitute \(x = 2y\)**:
\[ P(2y) = 3(2y)^3 - 2(2y)^2y - 13(2y)y^2 + 10y^3 \]

3. **Calculate \(P(2y)\)**:
- Calculate each term:
- \(3(2y)^3 = 3 \cdot 8y^3 = 24y^3\)
- \(-2(2y)^2y = -2 \cdot 4y^2 \cdot y = -8y^3\)
- \(-13(2y)y^2 = -26y^3\)
- \(10y^3 = 10y^3\)

4. **Combine the terms**:
\[ P(2y) = 24y^3 - 8y^3 - 26y^3 + 10y^3 \]
- Combine the coefficients:
\[ = (24 - 8 - 26 + 10)y^3 = 0y^3 = 0 \]

5. **Conclusion**:
The remainder when \(3x^3 - 2x^2y - 13xy^2 + 10y^3\) is divided by \(x - 2y\) is \(0\).

### Final Answer:
The remainder is \(0\).

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