`(2x - 32x^(3)) =` ?

Options

A. `2 (x-4) (x +4)`
B. `2x (1- 2x )^(2)`
C. `2x (1 + 2x)^(2)`
D. `2x (1 - 4 x) (1 + 4x)`

Answer

To factor the expression \(2x - 32x^3\), we can follow these steps:

### Step 1: Identify the common factor
The first step is to identify the common factor in both terms of the expression. We have \(2x\) and \(32x^3\). The common factor is \(2x\).

### Step 2: Factor out the common factor
Next, we factor out \(2x\) from the expression:
\[ 2x - 32x^3 = 2x(1 - 16x^2) \]

### Step 3: Recognize the difference of squares
Now, we notice that \(1 - 16x^2\) can be expressed as a difference of squares. We can rewrite \(16x^2\) as \((4x)^2\):
\[ 1 - 16x^2 = 1^2 - (4x)^2 \]

### Step 4: Apply the difference of squares formula
Using the difference of squares formula, \(a^2 - b^2 = (a + b)(a - b)\), we can factor \(1 - 16x^2\):
\[ 1 - 16x^2 = (1 + 4x)(1 - 4x) \]

### Step 5: Combine the factors
Now we can substitute this back into our expression:
\[ 2x(1 - 16x^2) = 2x(1 + 4x)(1 - 4x) \]

### Final Answer
Thus, the fully factored form of the expression \(2x - 32x^3\) is:
\[ 2x(1 + 4x)(1 - 4x) \]

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