If `m + n =1,` then the value of `m ^(3) + n ^(3) + 3 mn`

Options

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

Answer

To solve the problem, we need to find the value of \( m^3 + n^3 + 3mn \) given that \( m + n = 1 \).

### Step-by-Step Solution:

1. **Use the identity for the sum of cubes**:
We can use the identity:
\[ m^3 + n^3 + 3mn = (m+n)(m^2 - mn + n^2) \]
However, we can also use another identity:
\[ m^3 + n^3 = (m+n)(m^2 - mn + n^2) \]
This means we can express \( m^3 + n^3 + 3mn \) in terms of \( m+n \).

2. **Substitute \( m+n \)**:
Since we know \( m + n = 1 \), we can substitute this into the identity:
\[ m^3 + n^3 + 3mn = (m+n)((m+n)^2 - 3mn) \]
Substituting \( m+n = 1 \):
\[ m^3 + n^3 + 3mn = 1 \cdot (1^2 - 3mn) = 1 - 3mn \]

3. **Find \( mn \)**:
We need to express \( mn \) in terms of \( m+n \). We know that:
\[ (m+n)^2 = m^2 + 2mn + n^2 \]
Therefore:
\[ 1^2 = m^2 + n^2 + 2mn \]
This simplifies to:
\[ 1 = m^2 + n^2 + 2mn \]
We can express \( m^2 + n^2 \) using the identity:
\[ m^2 + n^2 = (m+n)^2 - 2mn = 1 - 2mn \]
Substituting this back gives:
\[ 1 = (1 - 2mn) + 2mn \]
This means:
\[ 1 = 1 \]
which is always true, indicating we need more information about \( mn \).

4. **Conclusion**:
Since we have \( m+n = 1 \) and we cannot determine \( mn \) without additional information, we can conclude:
\[ m^3 + n^3 + 3mn = 1 - 3mn \]
However, if we assume \( mn = 0 \) (which is true if either \( m \) or \( n \) is 0), we can find:
\[ m^3 + n^3 + 3mn = 1 - 0 = 1 \]
Therefore, the value of \( m^3 + n^3 + 3mn \) is \( 1 \).

### Final Answer:
The value of \( m^3 + n^3 + 3mn \) is \( 1 \).