The factors of `(x^(4)+16)` are :

Options

A. `(x^(2)+4)(x^(2)-4)`
B. `(x^(2)+4)(x^(2)+4)`
C. `x(x^(2)+2)(x^(4)+4)`
D. do not exist.

Answer

To factor the expression \( x^4 + 16 \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ x^4 + 16 \]
We can rewrite \( 16 \) as \( 4^2 \):
\[ x^4 + 4^2 \]

### Step 2: Introduce a Middle Term
To factor this expression, we can add and subtract a term. We can add \( 8x^2 \) and subtract \( 8x^2 \):
\[ x^4 + 8x^2 - 8x^2 + 16 \]
This does not change the value of the expression.

### Step 3: Group the Terms
Now we can group the terms:
\[ (x^4 + 8x^2 + 16) - 8x^2 \]

### Step 4: Recognize a Perfect Square
The first group \( x^4 + 8x^2 + 16 \) can be recognized as a perfect square:
\[ (x^2 + 4)^2 \]
Thus, we rewrite the expression:
\[ (x^2 + 4)^2 - (2\sqrt{2}x)^2 \]

### Step 5: Apply the Difference of Squares
Now we can apply the difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \):
Let \( a = (x^2 + 4) \) and \( b = 2\sqrt{2}x \):
\[ (x^2 + 4 + 2\sqrt{2}x)(x^2 + 4 - 2\sqrt{2}x) \]

### Step 6: Write the Final Factors
Thus, the factors of \( x^4 + 16 \) are:
\[ (x^2 + 4 + 2\sqrt{2}x)(x^2 + 4 - 2\sqrt{2}x) \]

### Summary
The factors of \( x^4 + 16 \) are:
\[ (x^2 + 4 + 2\sqrt{2}x)(x^2 + 4 - 2\sqrt{2}x) \]