The factor form `8-4x-2x^(3)+x^(4)` is

Options

A. `(2-x)(4-x^(3))`
B. `(2+x)(4-x^(3))`
C. `(2+x)(4+x^(3))`
D. `(2-x)(4+x^(3))`

Answer

To factor the polynomial \( 8 - 4x - 2x^3 + x^4 \), we can follow these steps:

### Step 1: Rearrange the terms
Rearranging the terms can sometimes make it easier to see common factors. We can rewrite the polynomial as:
\[ x^4 - 2x^3 - 4x + 8 \]

### Step 2: Group the terms
Next, we can group the terms in pairs:
\[ (x^4 - 2x^3) + (-4x + 8) \]

### Step 3: Factor out the common factors in each group
Now, we can factor out the common factors from each group:
- From the first group \( x^4 - 2x^3 \), we can factor out \( 2x^3 \):
\[ x^3(x - 2) \]
- From the second group \( -4x + 8 \), we can factor out \(-4\):
\[ -4(x - 2) \]

So, we rewrite the expression as:
\[ x^3(x - 2) - 4(x - 2) \]

### Step 4: Factor out the common binomial factor
Now we can see that \( (x - 2) \) is a common factor:
\[ (x - 2)(x^3 - 4) \]

### Step 5: Factor the remaining cubic expression
The remaining expression \( x^3 - 4 \) can be factored as a difference of cubes:
\[ x^3 - 4 = x^3 - 2^2 \]
This can be factored using the formula \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \):
Here, \( a = x \) and \( b = 2 \):
\[ x^3 - 4 = (x - 2)(x^2 + 2x + 4) \]

### Step 6: Combine the factors
Now, substituting back, we have:
\[ (x - 2)(x - 2)(x^2 + 2x + 4) \]
or
\[ (x - 2)^2(x^2 + 2x + 4) \]

### Final Answer
Thus, the factor form of \( 8 - 4x - 2x^3 + x^4 \) is:
\[ (x - 2)^2(x^2 + 2x + 4) \]