Final answer:
To factor the polynomial (x^8 − 6561) completely, recognize 6561 is 81^2 and use the difference of squares strategy. This results in the factored form (x^4 + 81)(x^2 + 9)(x + 3)(x - 3).
Step-by-step explanation:
The student's question involves factoring the polynomial expression (x^8 − 6561). To factor this expression completely, we use the difference of squares method. Recognize that 6561 is a perfect square, specifically 81^2. Hence, the expression can be rewritten as (x^4)^2 − 81^2, which is a difference of perfect squares and can be factored as (x^4 + 81)(x^4 - 81).
Now we have to factor (x^4 - 81) further since it is also a difference of squares: (x^2)^2 − 9^2, which can be factored as (x^2 + 9)(x^2 - 9). Again, we have another difference of squares for (x^2 - 9), factoring to (x + 3)(x - 3). Combining all factors, the completely factored form of the original expression is (x^4 + 81)(x^2 + 9)(x + 3)(x - 3).