If y+2m is a factor of `y^(5)-4m^(2)y^(3)+2y+2m+3` then value of is-

Options

A. `2/3`
B. `3/2`
C. 1
D. `(-3)/2`

Answer

To solve the problem, we will use the Factor Theorem, which states that if \( y + 2m \) is a factor of the polynomial \( P(y) = y^5 - 4m^2y^3 + 2y + 2m + 3 \), then substituting \( y = -2m \) into the polynomial should yield zero.

### Step-by-Step Solution:

1. **Identify the Polynomial**:
\[ P(y) = y^5 - 4m^2y^3 + 2y + 2m + 3 \]

2. **Set the Factor Equal to Zero**:
\[ y + 2m = 0 \implies y = -2m \]

3. **Substitute \( y = -2m \) into the Polynomial**:
\[ P(-2m) = (-2m)^5 - 4m^2(-2m)^3 + 2(-2m) + 2m + 3 \]

4. **Calculate Each Term**:
- First term:
\[ (-2m)^5 = -32m^5 \]
- Second term:
\[ -4m^2(-2m)^3 = -4m^2(-8m^3) = 32m^5 \]
- Third term:
\[ 2(-2m) = -4m \]
- Fourth term:
\[ 2m \]
- Fifth term:
\[ 3 \]

5. **Combine All Terms**:
\[ P(-2m) = -32m^5 + 32m^5 - 4m + 2m + 3 \]
Simplifying this gives:
\[ P(-2m) = 0 - 2m + 3 = -2m + 3 \]

6. **Set the Polynomial Equal to Zero**:
\[ -2m + 3 = 0 \]

7. **Solve for \( m \)**:
\[ -2m = -3 \implies m = \frac{3}{2} \]

### Final Answer:
The value of \( m \) is \( \frac{3}{2} \).

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