Factorise : `16(a-b)^(3)-24(a-b)^(2)` . The factors are :

Options

A. `4(a-b)(2a-b-3)`
B. `8(a-b)^(2)(2a-2b-3)`
C. `8(a-b)(2a-b-4)`
D. None of these

Answer

To factorise the expression \( 16(a-b)^3 - 24(a-b)^2 \), we can follow these steps:

### Step 1: Identify the common factor
We notice that both terms in the expression \( 16(a-b)^3 \) and \( -24(a-b)^2 \) have a common factor of \( (a-b)^2 \).

### Step 2: Factor out the common factor
We can factor out \( (a-b)^2 \) from both terms:
\[ 16(a-b)^3 - 24(a-b)^2 = (a-b)^2 [16(a-b) - 24] \]

### Step 3: Simplify the expression inside the brackets
Now, we simplify the expression inside the brackets:
\[ 16(a-b) - 24 = 16a - 16b - 24 \]

### Step 4: Factor out the numerical coefficient
Next, we can factor out the common numerical coefficient from \( 16a - 16b - 24 \):
\[ 16a - 16b - 24 = 8(2a - 2b - 3) \]

### Step 5: Combine the factors
Now, we can combine our factors:
\[ 16(a-b)^3 - 24(a-b)^2 = (a-b)^2 \cdot 8(2a - 2b - 3) \]

### Final Factorised Form
Thus, the final factorised form of the expression is:
\[ 8(a-b)^2(2a - 2b - 3) \]