Factor.

Question

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: \[ \boxed{2x(x - 4)(x + 4)(x^2 + 16)} \]