A common factor of `x^(4) -256, x^(3)-4x^(2)+3x-12 and x^(2) -7x+12` is

Options

A. `x+4`
B. `x-3`
C. `x-4`
D. `x+3`

Answer

To find a common factor of the expressions \( x^4 - 256 \), \( x^3 - 4x^2 + 3x - 12 \), and \( x^2 - 7x + 12 \), we will factor each expression step by step.

### Step 1: Factor \( x^4 - 256 \)

1. Recognize that \( x^4 - 256 \) can be rewritten as \( x^4 - 16^2 \).
2. This is a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \).
3. Thus, we can factor it as:
\[ x^4 - 256 = (x^2 - 16)(x^2 + 16) \]
4. The term \( x^2 - 16 \) can be further factored as:
\[ x^2 - 16 = (x - 4)(x + 4) \]
5. Therefore, the complete factorization of \( x^4 - 256 \) is:
\[ x^4 - 256 = (x - 4)(x + 4)(x^2 + 16) \]

### Step 2: Factor \( x^3 - 4x^2 + 3x - 12 \)

1. Group the terms:
\[ (x^3 - 4x^2) + (3x - 12) \]
2. Factor out the common terms in each group:
\[ x^2(x - 4) + 3(x - 4) \]
3. Now, factor out the common factor \( (x - 4) \):
\[ (x - 4)(x^2 + 3) \]

### Step 3: Factor \( x^2 - 7x + 12 \)

1. Look for two numbers that multiply to \( 12 \) and add to \( -7 \). These numbers are \( -3 \) and \( -4 \).
2. Thus, we can factor it as:
\[ x^2 - 7x + 12 = (x - 3)(x - 4) \]

### Step 4: Identify the Common Factor

Now we have the factorizations:
- From \( x^4 - 256 \): \( (x - 4)(x + 4)(x^2 + 16) \)
- From \( x^3 - 4x^2 + 3x - 12 \): \( (x - 4)(x^2 + 3) \)
- From \( x^2 - 7x + 12 \): \( (x - 3)(x - 4) \)

The common factor in all three expressions is \( (x - 4) \).

### Final Answer

The common factor of the given expressions is:
\[ \boxed{x - 4} \]