To find the factors of the expression x³ – 729, we’ll first recognize that 729 is a perfect cube. Specifically, 729 can be expressed as 9³ or (3²)³, which simplifies to 3^6. Therefore, we can rewrite the expression as x³ – 3³.
This matches the formula for the difference of cubes, which states that: a³ – b³ = (a – b)(a² + ab + b²).
In our case, we have:
- a = x
- b = 9
Applying the difference of cubes formula, we can factor x³ – 729 as follows:
x³ – 729 = (x – 9)(x² + 9x + 81)
Thus, one of the factors of x³ – 729 is (x – 9). This shows that if you substitute x = 9 into the equation, the result will equal zero, confirming that (x – 9) is a factor.