Factorise `(a+b)^(3)-a-b`.

Options

A. `(a+b)(a+b+1)(a-b-1)`
B. `(a+b)(a+b+1)(a+b-1)`
C. `(a+b)(a-b+1)(a+b-1)`
D. `(a-b)(a+b+1)(a+b-1)`

Answer

To factorize the expression \((a+b)^{3} - a - b\), we can follow these steps:

### Step 1: Rewrite the Expression
Start with the given expression:
\[ (a+b)^{3} - a - b \]

### Step 2: Group Terms
We can group the last two terms:
\[ (a+b)^{3} - (a + b) \]

### Step 3: Factor Out Common Terms
Notice that both terms contain \((a+b)\). We can factor \((a+b)\) out:
\[ = (a+b)^{3} - 1(a+b) \]
This can be rewritten as:
\[ = (a+b)\left((a+b)^{2} - 1\right) \]

### Step 4: Recognize the Difference of Squares
The expression \((a+b)^{2} - 1\) is a difference of squares, which can be factored using the identity \(x^{2} - y^{2} = (x+y)(x-y)\):
\[ = (a+b)\left((a+b) + 1\right)\left((a+b) - 1\right) \]

### Step 5: Final Factorization
Thus, the fully factored form of the expression is:
\[ = (a+b)(a+b+1)(a+b-1) \]

### Final Answer
The factorization of \((a+b)^{3} - a - b\) is:
\[ (a+b)(a+b+1)(a+b-1) \]

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