What is the simplified form of the expression $$ (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 - x) $$?

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**Solution:**

1. Start with the given expression:
   
   $$ (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 - x) $$

2. Group the first and last terms:
   
   $$ (1 + x)(1 - x) = 1 - x^2 $$

3. Substitute back:
   
   $$ (1 - x^2)(1 + x^2)(1 + x^4)(1 + x^8) $$

4. Next, multiply the first two factors:
   
   $$ (1 - x^2)(1 + x^2) = 1 - x^4 $$

5. Substitute back:
   
   $$ (1 - x^4)(1 + x^4)(1 + x^8) $$

6. Multiply the first two factors again:
   
   $$ (1 - x^4)(1 + x^4) = 1 - x^8 $$

7. Substitute back:
   
   $$ (1 - x^8)(1 + x^8) $$

8. Finally, multiply these two:
   
   $$ (1 - x^8)(1 + x^8) = 1 - x^{16} $$

**Answer:** $$1 - x^{16}$$

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