The factors of `x ^(8) - x ^(4) - 30` are :

Options

A. `(x ^(4) - 6) and (x ^(4) - 5)`
B. `(x ^(4) - 6) and (x ^(4) + 5)`
C. `(x ^(4) + 6) and (x ^(4) -5)`
D. `(x ^(4) + 6) and (x ^(4) +5)`

Answer

To factor the expression \( x^8 - x^4 - 30 \), we can follow these steps:

### Step 1: Substitute \( y = x^4 \)
We start by substituting \( y \) for \( x^4 \). This simplifies our expression:
\[ x^8 - x^4 - 30 = y^2 - y - 30 \]

### Step 2: Factor the quadratic expression
Next, we need to factor the quadratic expression \( y^2 - y - 30 \). We look for two numbers that multiply to \(-30\) (the constant term) and add up to \(-1\) (the coefficient of \(y\)).

The numbers that satisfy this are \( -6 \) and \( 5 \):
- Product: \( -6 \times 5 = -30 \)
- Sum: \( -6 + 5 = -1 \)

### Step 3: Rewrite the quadratic expression
Using these factors, we can rewrite the quadratic:
\[ y^2 - y - 30 = (y - 6)(y + 5) \]

### Step 4: Substitute back \( y = x^4 \)
Now we substitute back \( y = x^4 \):
\[ (x^4 - 6)(x^4 + 5) \]

### Step 5: Write the final factors
Thus, the factors of the original expression \( x^8 - x^4 - 30 \) are:
\[ (x^4 - 6)(x^4 + 5) \]

### Final Answer
The factors of \( x^8 - x^4 - 30 \) are \( x^4 - 6 \) and \( x^4 + 5 \).

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