Find the expansion of `(1+2x−x^2)^4`

Answer

To find the expansion of \( (1 + 2x - x^2)^4 \) using the Binomial Theorem, we can follow these steps:

### Step 1: Identify the terms for the Binomial Expansion
We can treat \( (1 + 2x - x^2) \) as a single term. Let's denote \( y = 2x - x^2 \). Thus, we rewrite the expression as \( (1 + y)^4 \).

### Step 2: Apply the Binomial Theorem
According to the Binomial Theorem, the expansion of \( (a + b)^n \) is given by:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, \( a = 1 \), \( b = y = 2x - x^2 \), and \( n = 4 \). Therefore, we have:
\[ (1 + (2x - x^2))^4 = \sum_{k=0}^{4} \binom{4}{k} (1)^{4-k} (2x - x^2)^k \]

### Step 3: Calculate each term in the expansion
Now we need to calculate each term in the sum for \( k = 0, 1, 2, 3, 4 \):

1. **For \( k = 0 \)**:
\[ \binom{4}{0} (1)^{4} (2x - x^2)^0 = 1 \]

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

3. **For \( k = 2 \)**:
\[ \binom{4}{2} (1)^{2} (2x - x^2)^2 = 6(2x - x^2)^2 \]
Expanding \( (2x - x^2)^2 \):
\[ (2x - x^2)^2 = 4x^2 - 4x^3 + x^4 \]
Thus,
\[ 6(4x^2 - 4x^3 + x^4) = 24x^2 - 24x^3 + 6x^4 \]

4. **For \( k = 3 \)**:
\[ \binom{4}{3} (1)^{1} (2x - x^2)^3 = 4(2x - x^2)^3 \]
Expanding \( (2x - x^2)^3 \):
\[ (2x - x^2)^3 = 8x^3 - 12x^4 + 6x^5 - x^6 \]
Thus,
\[ 4(8x^3 - 12x^4 + 6x^5 - x^6) = 32x^3 - 48x^4 + 24x^5 - 4x^6 \]

5. **For \( k = 4 \)**:
\[ \binom{4}{4} (1)^{0} (2x - x^2)^4 = (2x - x^2)^4 \]
Expanding \( (2x - x^2)^4 \):
\[ (2x - x^2)^4 = 16x^4 - 32x^5 + 24x^6 - 8x^7 + x^8 \]

### Step 4: Combine all terms
Now we combine all the terms we calculated:
\[ 1 + (8x - 4x^2) + (24x^2 - 24x^3 + 6x^4) + (32x^3 - 48x^4 + 24x^5 - 4x^6) + (16x^4 - 32x^5 + 24x^6 - 8x^7 + x^8) \]

Combining like terms:
- Constant term: \( 1 \)
- Coefficient of \( x \): \( 8x \)
- Coefficient of \( x^2 \): \( -4x^2 + 24x^2 = 20x^2 \)
- Coefficient of \( x^3 \): \( -24x^3 + 32x^3 = 8x^3 \)
- Coefficient of \( x^4 \): \( 6x^4 - 48x^4 + 16x^4 = -26x^4 \)
- Coefficient of \( x^5 \): \( 24x^5 - 32x^5 = -8x^5 \)
- Coefficient of \( x^6 \): \( -4x^6 + 24x^6 = 20x^6 \)
- Coefficient of \( x^7 \): \( -8x^7 \)
- Coefficient of \( x^8 \): \( x^8 \)

### Final Expansion
Thus, the final expansion of \( (1 + 2x - x^2)^4 \) is:
\[ 1 + 8x + 20x^2 + 8x^3 - 26x^4 - 8x^5 + 20x^6 - 8x^7 + x^8 \]