A complete factorisation of `( x ^(4) + 64)` is

Options

A. A)`( x ^(2) + 8) ^(2)`
B. B)` ( x^(2) + 8) ( x^(2) - 8)`
C. C)` ( x^(2) - 4 x + 8) ( x^(2) - 4 x - 8)`
D. D)` ( x^(2) + 4 x + 8 ) ( x^(2) - 4 x + 8)`

Answer

To completely factor the expression \( x^4 + 64 \), we can follow these steps:

### Step 1: Recognize the expression as a sum of squares
The expression \( x^4 + 64 \) can be rewritten as:
\[ x^4 + (8)^2 \]
This is a sum of squares, which can be factored using the identity for the sum of two squares.

### Step 2: Apply the sum of squares formula
We can use the identity:
\[ a^4 + b^4 = (a^2 + b^2 - ab)(a^2 + b^2 + ab) \]
In our case, let \( a = x^2 \) and \( b = 8 \). Then we have:
\[ x^4 + 64 = (x^2)^2 + (8)^2 \]

### Step 3: Calculate \( a^2 + b^2 \) and \( ab \)
First, calculate \( a^2 + b^2 \):
\[ a^2 + b^2 = (x^2)^2 + 8^2 = x^4 + 64 \]
Next, calculate \( ab \):
\[ ab = x^2 \cdot 8 = 8x^2 \]

### Step 4: Substitute into the sum of squares formula
Now, substituting back into the formula gives us:
\[ (x^2 + 8 - 8x)(x^2 + 8 + 8x) \]

### Step 5: Simplify the factors
Now we simplify the factors:
1. \( x^2 - 8x + 8 \)
2. \( x^2 + 8x + 8 \)

Thus, the complete factorization of \( x^4 + 64 \) is:
\[ (x^2 - 8x + 8)(x^2 + 8x + 8) \]

### Final Answer
The complete factorization of \( x^4 + 64 \) is:
\[ (x^2 - 8x + 8)(x^2 + 8x + 8) \]

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