The coefficient of `x` in the expansion of `(x+3)^(3)`is

Options

A. 1
B. 9
C. 18
D. 27

Explanation

To find the coefficient of \( x \) in the expansion of \( (x + 3)^3 \), we can use the binomial theorem or the polynomial identity for the cube of a binomial. Here’s the step-by-step solution:

### Step 1: Identify the expression
We need to expand \( (x + 3)^3 \).

### Step 2: Use the binomial expansion formula
The binomial expansion of \( (a + b)^n \) is given by:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For our case, \( a = x \), \( b = 3 \), and \( n = 3 \).

### Step 3: Write the expansion
Using the formula, we can expand \( (x + 3)^3 \):
\[ (x + 3)^3 = \binom{3}{0} x^3 \cdot 3^0 + \binom{3}{1} x^2 \cdot 3^1 + \binom{3}{2} x^1 \cdot 3^2 + \binom{3}{3} x^0 \cdot 3^3 \]

### Step 4: Calculate the coefficients
Now, we calculate each term:
1. \( \binom{3}{0} x^3 \cdot 3^0 = 1 \cdot x^3 \cdot 1 = x^3 \)
2. \( \binom{3}{1} x^2 \cdot 3^1 = 3 \cdot x^2 \cdot 3 = 9x^2 \)
3. \( \binom{3}{2} x^1 \cdot 3^2 = 3 \cdot x \cdot 9 = 27x \)
4. \( \binom{3}{3} x^0 \cdot 3^3 = 1 \cdot 1 \cdot 27 = 27 \)

### Step 5: Combine the terms
Combining all the terms, we get:
\[ (x + 3)^3 = x^3 + 9x^2 + 27x + 27 \]

### Step 6: Identify the coefficient of \( x \)
From the expanded expression, the coefficient of \( x \) is \( 27 \).

### Final Answer
The coefficient of \( x \) in the expansion of \( (x + 3)^3 \) is \( 27 \).

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