`x ^(3) - x =? `

Options

A. `x (x ^(2) -x)`
B. `x (x -x ^(2))`
C. `x (1+x) (1 -x)`
D. `x (x +1) (x -1)`

Answer

To factor the expression \( x^3 - x \), we can follow these steps:

### Step 1: Identify the common factor
The expression \( x^3 - x \) has a common factor of \( x \).

### Step 2: Factor out the common factor
We can factor out \( x \) from the expression:
\[ x^3 - x = x(x^2 - 1) \]

### Step 3: Recognize the difference of squares
The expression \( x^2 - 1 \) is a difference of squares, which can be factored further. We know that:
\[ a^2 - b^2 = (a + b)(a - b) \]
In our case, \( a = x \) and \( b = 1 \).

### Step 4: Factor the difference of squares
Applying the difference of squares formula, we can factor \( x^2 - 1 \):
\[ x^2 - 1 = (x + 1)(x - 1) \]

### Step 5: Combine the factors
Now we can combine all the factors together:
\[ x^3 - x = x(x^2 - 1) = x(x + 1)(x - 1) \]

### Final Answer
Thus, the fully factored form of \( x^3 - x \) is:
\[ x(x + 1)(x - 1) \]

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