Question

Factor.
2 x Superscript 5 Baseline minus 512 x
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Part 1
2 x Superscript 5 Baseline minus 512 xequals
  
enter your response here ​(Factor completely.)

Answer 1

To factor the expression \(2x^5 - 512x\), we can first look for a common factor.

1. **Identify the common factor**: Both terms share a factor of \(2x\).

2. **Factor out \(2x\)**:
\[
2x^5 - 512x = 2x(x^4 - 256)
\]

3. **Notice that \(256\) is a perfect square**:
\[
256 = 16^2
\]

Thus, the expression \(x^4 - 256\) can be recognized as a difference of squares:
\[
x^4 - 256 = (x^2)^2 - (16)^2
\]

4. **Apply the difference of squares factorization**:
\[
x^4 - 256 = (x^2 - 16)(x^2 + 16)
\]

5. **The term \(x^2 - 16\) is also a difference of squares**:
\[
x^2 - 16 = (x - 4)(x + 4)
\]

6. **Now we can write the complete factorization**:
Combining all the factors, we get:
\[
2x(x^2 - 16)(x^2 + 16) = 2x(x - 4)(x + 4)(x^2 + 16)
\]

So the completely factored form of the original expression \(2x^5 - 512x\) is:
\[
\boxed{2x(x - 4)(x + 4)(x^2 + 16)}
\]