The value of lambda for which the expression `x^(3) + x^(2) - 5x + lambda` will be divisible by (x-2) is :

Options

A. 2
B. `-2`
C. `-3`
D. 4

Answer

To find the value of lambda for which the expression \( x^3 + x^2 - 5x + \lambda \) is divisible by \( x - 2 \), we can use the Factor Theorem. According to the Factor Theorem, if a polynomial \( f(x) \) is divisible by \( x - c \), then \( f(c) = 0 \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
The polynomial is \( f(x) = x^3 + x^2 - 5x + \lambda \) and the divisor is \( x - 2 \).

2. **Apply the Factor Theorem**:
For \( f(x) \) to be divisible by \( x - 2 \), we need to find \( f(2) \) and set it equal to 0.

3. **Substitute \( x = 2 \) into the polynomial**:
\[ f(2) = (2)^3 + (2)^2 - 5(2) + \lambda \]

4. **Calculate \( f(2) \)**:
\[ f(2) = 8 + 4 - 10 + \lambda \]
Simplifying this gives:
\[ f(2) = 2 + \lambda \]

5. **Set \( f(2) \) equal to 0**:
\[ 2 + \lambda = 0 \]

6. **Solve for \( \lambda \)**:
\[ \lambda = -2 \]

### Final Answer:
The value of \( \lambda \) for which the expression \( x^3 + x^2 - 5x + \lambda \) is divisible by \( x - 2 \) is \( \lambda = -2 \).

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