Factors of `x^(4)-x^(2)-12` are

Options

A. `(x+2), (x-2), (x^(2)+3)`
B. `(x+3), (x-3), (x^(2)+2)`
C. `(x+2), (x-2), (x^(2)-3)`
D. `(x^(2)+2),(x^(2)-6)`

Answer

To factor the polynomial \( x^4 - x^2 - 12 \), we can follow these steps:

### Step 1: Substitute \( y = x^2 \)
We can simplify the polynomial by substituting \( y \) for \( x^2 \). This gives us:
\[ y^2 - y - 12 \]

### Step 2: Factor the quadratic
Next, we need to factor the quadratic expression \( y^2 - y - 12 \). We are looking for two numbers that multiply to \(-12\) (the constant term) and add to \(-1\) (the coefficient of \(y\)).

The numbers that satisfy this are \( -4 \) and \( 3 \) because:
\[ -4 \times 3 = -12 \quad \text{and} \quad -4 + 3 = -1 \]

Thus, we can factor the quadratic as:
\[ (y - 4)(y + 3) \]

### Step 3: Substitute back \( x^2 \)
Now we substitute back \( y = x^2 \) into the factored form:
\[ (x^2 - 4)(x^2 + 3) \]

### Step 4: Factor further if possible
The term \( x^2 - 4 \) can be factored further using the difference of squares:
\[ x^2 - 4 = (x - 2)(x + 2) \]

So, the complete factorization of the polynomial \( x^4 - x^2 - 12 \) is:
\[ (x - 2)(x + 2)(x^2 + 3) \]

### Final Answer
The factors of \( x^4 - x^2 - 12 \) are:
\[ (x - 2)(x + 2)(x^2 + 3) \]

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