Find the value of l, so that `y-2p` is a factor of `(y^(3))/(4p^(2))-2y+lp` .

Options

A. `0`
B. `1`
C. `2`
D. `3`

Answer

To find the value of \( l \) such that \( y - 2p \) is a factor of the polynomial

\[ \frac{y^3}{4p^2} - 2y + lp, \]

we can use the factor theorem. According to the factor theorem, if \( y - 2p \) is a factor of the polynomial, then substituting \( y = 2p \) into the polynomial should yield a result of zero.

### Step-by-step Solution:

1. **Substitute \( y = 2p \) into the polynomial**:
\[ \frac{(2p)^3}{4p^2} - 2(2p) + lp. \]

2. **Calculate \( (2p)^3 \)**:
\[ (2p)^3 = 8p^3. \]

3. **Substitute this back into the polynomial**:
\[ \frac{8p^3}{4p^2} - 4p + lp. \]

4. **Simplify \( \frac{8p^3}{4p^2} \)**:
\[ \frac{8p^3}{4p^2} = 2p. \]

5. **Now, substitute this into the polynomial**:
\[ 2p - 4p + lp. \]

6. **Combine like terms**:
\[ (2p - 4p + lp) = (-2p + lp). \]

7. **Set the polynomial equal to zero (since \( y - 2p \) is a factor)**:
\[ -2p + lp = 0. \]

8. **Rearrange the equation to solve for \( l \)**:
\[ lp = 2p. \]

9. **Divide both sides by \( p \) (assuming \( p \neq 0 \))**:
\[ l = 2. \]

### Final Answer:
The value of \( l \) is \( 2 \).

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