The polynomial `x^(4)+12x^(2)+64` on factorization gives ___________.

Options

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

Answer

To factorize the polynomial \(x^4 + 12x^2 + 64\), we can follow these steps:

### Step 1: Rewrite the Polynomial
We start with the polynomial:
\[ x^4 + 12x^2 + 64 \]

### Step 2: Substitute a Variable
Let’s substitute \(y = x^2\). This transforms our polynomial into:
\[ y^2 + 12y + 64 \]

### Step 3: Factor the Quadratic
Now, we need to factor the quadratic \(y^2 + 12y + 64\). We look for two numbers that multiply to \(64\) (the constant term) and add up to \(12\) (the coefficient of \(y\)). The numbers \(8\) and \(4\) satisfy this condition:
\[ y^2 + 8y + 4y + 64 = (y + 8)(y + 4) \]

### Step 4: Substitute Back
Now, we substitute back \(y = x^2\):
\[ (x^2 + 8)(x^2 + 4) \]

### Step 5: Final Factorization
Thus, the factorization of the original polynomial \(x^4 + 12x^2 + 64\) is:
\[ (x^2 + 8)(x^2 + 4) \]

### Conclusion
The polynomial \(x^4 + 12x^2 + 64\) can be factored as:
\[ (x^2 + 8)(x^2 + 4) \]

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