The polynomial `x^(3) -4x^(2) + x -4` on factorization gives

Options

A. `(x -4) (x^(2) -1)`
B. `(x -4) (x^(2) +4)`
C. `(x +4) (x^(2) +1)`
D. `(x -4) (x^(2) +1)`

Explanation

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

### Step 1: Group the terms
We can group the terms of the polynomial to make it easier to factor. We will group the first two terms and the last two terms:
\[ (x^3 - 4x^2) + (x - 4) \]

### Step 2: Factor out the common terms in each group
Now, we can factor out the common terms from each group:
- From the first group \( x^3 - 4x^2 \), we can factor out \( x^2 \):
\[ x^2(x - 4) \]
- From the second group \( x - 4 \), we can factor out \( 1 \):
\[ 1(x - 4) \]

Now we rewrite the expression:
\[ x^2(x - 4) + 1(x - 4) \]

### Step 3: Factor out the common binomial factor
Now we can see that \( (x - 4) \) is a common factor:
\[ (x - 4)(x^2 + 1) \]

### Final Answer
Thus, the factorization of the polynomial \( x^3 - 4x^2 + x - 4 \) is:
\[ (x - 4)(x^2 + 1) \]