The LCM and HCF of two polynomials are respectively `(2a-5)^2 (a + 1)` and `(2a-5)`. If one of the polynomials is `4a^2 – 20a +25`, the other one is :

Options

A. `4a^2 +20a+5`
B. `4a^2 – 25`
C. `2a^2 +3a-5`
D. `2a^2 – 3a - 5 `

Answer

To find the other polynomial given the LCM and HCF of two polynomials, we can use the relationship:

\[ \text{LCM} \times \text{HCF} = P_1 \times P_2 \]

Where:
- LCM is the least common multiple of the two polynomials,
- HCF is the highest common factor of the two polynomials,
- \( P_1 \) is one polynomial (given as \( 4a^2 - 20a + 25 \)),
- \( P_2 \) is the other polynomial (which we need to find).

### Step-by-step Solution:

1. **Identify the Given Values**:
- LCM = \( (2a - 5)^2 (a + 1) \)
- HCF = \( (2a - 5) \)
- \( P_1 = 4a^2 - 20a + 25 \)

2. **Calculate the Product of LCM and HCF**:
\[ \text{LCM} \times \text{HCF} = (2a - 5)^2 (a + 1) \times (2a - 5) \]
This simplifies to:
\[ = (2a - 5)^3 (a + 1) \]

3. **Express the Relationship**:
According to the relationship:
\[ (2a - 5)^3 (a + 1) = P_1 \times P_2 \]

4. **Substitute \( P_1 \)**:
Substitute \( P_1 \) into the equation:
\[ (2a - 5)^3 (a + 1) = (4a^2 - 20a + 25) \times P_2 \]

5. **Factor \( P_1 \)**:
Factor \( P_1 \):
\[ 4a^2 - 20a + 25 = (2a - 5)^2 \]
So, we can rewrite the equation:
\[ (2a - 5)^3 (a + 1) = (2a - 5)^2 \times P_2 \]

6. **Divide Both Sides by \( (2a - 5)^2 \)**:
\[ (2a - 5)(a + 1) = P_2 \]

7. **Expand \( P_2 \)**:
Now, expand \( P_2 \):
\[ P_2 = (2a - 5)(a + 1) = 2a^2 + 2a - 5a - 5 = 2a^2 - 3a - 5 \]

8. **Final Result**:
Therefore, the other polynomial \( P_2 \) is:
\[ P_2 = 2a^2 - 3a - 5 \]