Using binomial theorem, expand each of the following:`(1+2x-3x^(2))^(4)`

Explanation

To expand the expression \((1 + 2x - 3x^2)^4\) using the Binomial Theorem, we can follow these steps:

### Step 1: Identify the Binomial Form
We can rewrite the expression in a suitable form for the Binomial Theorem:
\[ (1 + (2x - 3x^2))^4 \]
Here, we can let \(y = 2x - 3x^2\).

### Step 2: Apply the Binomial Theorem
According to the Binomial Theorem, \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In our case:
- \(a = 1\)
- \(b = y = 2x - 3x^2\)
- \(n = 4\)

Thus, we can expand it as:
\[ (1 + y)^4 = \sum_{k=0}^{4} \binom{4}{k} (1)^{4-k} (2x - 3x^2)^k \]

### Step 3: Calculate Each Term
Now, we will calculate each term for \(k = 0\) to \(k = 4\):

1. **For \(k = 0\)**:
\[ \binom{4}{0} (1)^{4} (2x - 3x^2)^{0} = 1 \]

2. **For \(k = 1\)**:
\[ \binom{4}{1} (1)^{3} (2x - 3x^2)^{1} = 4(2x - 3x^2) = 8x - 12x^2 \]

3. **For \(k = 2\)**:
\[ \binom{4}{2} (1)^{2} (2x - 3x^2)^{2} = 6(2x - 3x^2)^{2} \]
Expanding \((2x - 3x^2)^{2}\):
\[ (2x - 3x^2)(2x - 3x^2) = 4x^2 - 12x^3 + 9x^4 \]
Thus,
\[ 6(4x^2 - 12x^3 + 9x^4) = 24x^2 - 72x^3 + 54x^4 \]

4. **For \(k = 3\)**:
\[ \binom{4}{3} (1)^{1} (2x - 3x^2)^{3} = 4(2x - 3x^2)^{3} \]
Expanding \((2x - 3x^2)^{3}\):
\[ (2x - 3x^2)(2x - 3x^2)(2x - 3x^2) = 8x^3 - 36x^4 + 54x^5 - 27x^6 \]
Thus,
\[ 4(8x^3 - 36x^4 + 54x^5 - 27x^6) = 32x^3 - 144x^4 + 216x^5 - 108x^6 \]

5. **For \(k = 4\)**:
\[ \binom{4}{4} (1)^{0} (2x - 3x^2)^{4} = (2x - 3x^2)^{4} \]
Expanding \((2x - 3x^2)^{4}\) using the binomial theorem:
\[ = 16x^4 - 96x^5 + 216x^6 - 81x^8 \]

### Step 4: Combine All Terms
Now, we combine all the terms we calculated:
\[ 1 + (8x - 12x^2) + (24x^2 - 72x^3 + 54x^4) + (32x^3 - 144x^4 + 216x^5 - 108x^6) + (16x^4 - 96x^5 + 216x^6 - 81x^8) \]

Combining like terms:
- Constant term: \(1\)
- \(x\) term: \(8x\)
- \(x^2\) term: \(-12x^2 + 24x^2 = 12x^2\)
- \(x^3\) term: \(-72x^3 + 32x^3 = -40x^3\)
- \(x^4\) term: \(54x^4 - 144x^4 + 16x^4 = -74x^4\)
- \(x^5\) term: \(216x^5 - 96x^5 = 120x^5\)
- \(x^6\) term: \(-108x^6 + 216x^6 = 108x^6\)
- \(x^8\) term: \(-81x^8\)

Thus, the final expansion is:
\[ (1 + 2x - 3x^2)^4 = 81x^8 - 216x^7 + 108x^6 + 120x^5 - 74x^4 - 40x^3 + 12x^2 + 8x + 1 \]