Factors of `(a+b)^(3)-(a-b)^(3)` are

Options

A. `2ab,(3a^(2)+b^(2))`
B. `ab,(3a^(2)+b^(2))`
C. `2b,(3a^(2)+b^(2))`
D. `(3a^(2)+b^(2)), 2a`

Answer

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

### Step 1: Write down the expression
We start with the expression:
\[ (a+b)^{3} - (a-b)^{3} \]

### Step 2: Use the identities for cubes
We will use the identities for the cubes:
- The formula for \((x+y)^{3}\) is \(x^{3} + y^{3} + 3xy(x+y)\).
- The formula for \((x-y)^{3}\) is \(x^{3} - y^{3} - 3xy(x-y)\).

Here, let \(x = a\) and \(y = b\).

### Step 3: Expand \((a+b)^{3}\)
Using the identity for \((a+b)^{3}\):
\[ (a+b)^{3} = a^{3} + b^{3} + 3ab(a+b) \]

### Step 4: Expand \((a-b)^{3}\)
Using the identity for \((a-b)^{3}\):
\[ (a-b)^{3} = a^{3} - b^{3} - 3ab(a-b) \]

### Step 5: Substitute the expansions into the expression
Now substitute these expansions back into the original expression:
\[ (a+b)^{3} - (a-b)^{3} = \left(a^{3} + b^{3} + 3ab(a+b)\right) - \left(a^{3} - b^{3} - 3ab(a-b)\right) \]

### Step 6: Simplify the expression
Distributing the negative sign in the second part:
\[ = a^{3} + b^{3} + 3ab(a+b) - a^{3} + b^{3} + 3ab(a-b) \]
Now, combine like terms:
\[ = (a^{3} - a^{3}) + (b^{3} + b^{3}) + 3ab(a+b) + 3ab(a-b) \]
This simplifies to:
\[ = 2b^{3} + 3ab(a+b) + 3ab(a-b) \]

### Step 7: Combine the like terms
Now, we can combine the \(3ab\) terms:
\[ = 2b^{3} + 3ab(a+b + a-b) = 2b^{3} + 3ab(2a) \]
This gives us:
\[ = 2b^{3} + 6a b^{2} \]

### Step 8: Factor out common terms
Now, we can factor out the common terms:
\[ = 2b(b^{2} + 3a^{2}) \]

### Final Answer
Thus, the factors of \((a+b)^{3} - (a-b)^{3}\) are:
\[ 2b(3a^{2} + b^{2}) \]