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 follow these steps:

### Step 1: Understand the condition for divisibility
For a polynomial \( f(x) \) to be divisible by \( (x - a) \), it must satisfy \( f(a) = 0 \). In this case, since we want the polynomial to be divisible by \( (x - 2) \), we need to find \( f(2) \).

### Step 2: Define the polynomial
Let \( f(x) = x^3 + x^2 - 5x + \lambda \).

### Step 3: Substitute \( x = 2 \) into the polynomial
Now, we will substitute \( x = 2 \) into the polynomial:
\[ f(2) = 2^3 + 2^2 - 5 \cdot 2 + \lambda \]

### Step 4: Calculate \( f(2) \)
Calculating each term:
- \( 2^3 = 8 \)
- \( 2^2 = 4 \)
- \( -5 \cdot 2 = -10 \)

So,
\[ f(2) = 8 + 4 - 10 + \lambda \]
This simplifies to:
\[ f(2) = 2 + \lambda \]

### Step 5: Set \( f(2) = 0 \)
Since we need \( f(2) = 0 \) for the polynomial to be divisible by \( (x - 2) \), we set up the equation:
\[ 2 + \lambda = 0 \]

### Step 6: Solve for \( \lambda \)
Now, we can 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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