`a^(3)+3a^(2)b+3ab^(2)+b^(3)` divided by `a^(2)+2ab+b^(2)` is

Options

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

Answer

To solve the problem of dividing \( a^3 + 3a^2b + 3ab^2 + b^3 \) by \( a^2 + 2ab + b^2 \), we can follow these steps:

### Step 1: Identify the expressions
We have the numerator:
\[ a^3 + 3a^2b + 3ab^2 + b^3 \]
And the denominator:
\[ a^2 + 2ab + b^2 \]

### Step 2: Recognize the patterns
The numerator can be recognized as the expansion of \( (a + b)^3 \):
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]

The denominator can be recognized as the expansion of \( (a + b)^2 \):
\[ (a + b)^2 = a^2 + 2ab + b^2 \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression:
\[ \frac{a^3 + 3a^2b + 3ab^2 + b^3}{a^2 + 2ab + b^2} = \frac{(a + b)^3}{(a + b)^2} \]

### Step 4: Simplify the expression
Using the property of exponents that states \( \frac{x^m}{x^n} = x^{m-n} \), we can simplify:
\[ \frac{(a + b)^3}{(a + b)^2} = (a + b)^{3-2} = (a + b)^1 = a + b \]

### Final Answer
Thus, the result of the division is:
\[ a + b \]

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