Expand using binomial `(1+3 x)^3` upto the term having `x^3`.

Answer

To expand the expression \((1 + 3x)^3\) using the binomial theorem up to the term having \(x^3\), we can follow these steps:

### Step 1: Identify the components
In the expression \((1 + 3x)^3\), we have:
- \(a = 1\)
- \(b = 3x\)
- \(n = 3\)

### Step 2: Apply the binomial theorem
The binomial theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
where \(\binom{n}{k}\) is the binomial coefficient.

### Step 3: Write out the terms
We need to expand \((1 + 3x)^3\) using the binomial theorem:
\[ (1 + 3x)^3 = \sum_{k=0}^{3} \binom{3}{k} (1)^{3-k} (3x)^k \]

### Step 4: Calculate each term
Now we calculate the terms for \(k = 0\) to \(k = 3\):

1. **For \(k = 0\)**:
\[ \binom{3}{0} (1)^{3-0} (3x)^0 = 1 \cdot 1 \cdot 1 = 1 \]

2. **For \(k = 1\)**:
\[ \binom{3}{1} (1)^{3-1} (3x)^1 = 3 \cdot 1 \cdot 3x = 9x \]

3. **For \(k = 2\)**:
\[ \binom{3}{2} (1)^{3-2} (3x)^2 = 3 \cdot 1 \cdot (3x)^2 = 3 \cdot 9x^2 = 27x^2 \]

4. **For \(k = 3\)**:
\[ \binom{3}{3} (1)^{3-3} (3x)^3 = 1 \cdot 1 \cdot (27x^3) = 27x^3 \]

### Step 5: Combine the terms
Now we combine all the terms we calculated:
\[ (1 + 3x)^3 = 1 + 9x + 27x^2 + 27x^3 \]

### Final Result
Thus, the expansion of \((1 + 3x)^3\) up to the term having \(x^3\) is:
\[ 1 + 9x + 27x^2 + 27x^3 \]