Final answer:
To factor the expression 81x¹²-1 completely, first recognize it as a difference of squares and rewrite it as (9x⁶)² - (1)², which then factors into (9x⁶ + 1)(9x⁶ - 1). The factor 9x⁶ - 1 can be further factored, resulting in the final expression (9x⁶ + 1)(3x³ + 1)(3x³ - 1).
Step-by-step explanation:
To factor 81x¹²-1 completely, recognize that this expression is a difference of squares. A difference of squares is factored using the formula a² - b² = (a + b)(a - b). In this case, 81x¹² can be rewritten as (9x⁶)² and 1 as (1)². Therefore, the factored form of 81x¹²-1 is (9x⁶ + 1)(9x⁶ - 1).
However, notice that 9x⁶ - 1 can be further factored because it is, in itself, a difference of squares where a is 9x³ and b is 1. The completely factored form is:
(9x⁶ + 1)(3x³ + 1)(3x³ - 1)
Each factor represents a specific part of the original quadratic equation, and by setting them equal to zero and solving for x, you can find the roots of the original equation, if they exist within the real number system.