Using binomial theorem, expand each of the following: `(1-2x)^(5)`

Answer

To expand \((1 - 2x)^5\) using the Binomial Theorem, we can follow these steps:

### Step 1: Understand 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, which can be calculated as:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

### Step 2: Identify \(a\), \(b\), and \(n\)
In our case, we have:
- \(a = 1\)
- \(b = -2x\)
- \(n = 5\)

### Step 3: Write the Expansion
Using the Binomial Theorem, we can expand \((1 - 2x)^5\) as follows:
\[ (1 - 2x)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k} (-2x)^k \]

### Step 4: Calculate Each Term
Now we will calculate each term from \(k = 0\) to \(k = 5\):

1. For \(k = 0\):
\[ \binom{5}{0} (1)^{5-0} (-2x)^0 = 1 \cdot 1 \cdot 1 = 1 \]

2. For \(k = 1\):
\[ \binom{5}{1} (1)^{5-1} (-2x)^1 = 5 \cdot 1 \cdot (-2x) = -10x \]

3. For \(k = 2\):
\[ \binom{5}{2} (1)^{5-2} (-2x)^2 = 10 \cdot 1 \cdot 4x^2 = 40x^2 \]

4. For \(k = 3\):
\[ \binom{5}{3} (1)^{5-3} (-2x)^3 = 10 \cdot 1 \cdot (-8x^3) = -80x^3 \]

5. For \(k = 4\):
\[ \binom{5}{4} (1)^{5-4} (-2x)^4 = 5 \cdot 1 \cdot 16x^4 = 80x^4 \]

6. For \(k = 5\):
\[ \binom{5}{5} (1)^{5-5} (-2x)^5 = 1 \cdot 1 \cdot (-32x^5) = -32x^5 \]

### Step 5: Combine All Terms
Now, we combine all the terms we calculated:
\[ (1 - 2x)^5 = 1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5 \]

### Final Answer
Thus, the expansion of \((1 - 2x)^5\) is:
\[ 1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5 \]