Questions: Factor the binomial completely. s^3 - 729 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. s^3 - 729 = (Factor completely.) B. The binomial is prime.

Solution Steps

To factor the binomial \( s^3 - 729 \) completely, we recognize that it is a difference of cubes. The difference of cubes formula is given by: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] Here, \( s^3 - 729 \) can be written as \( s^3 - 9^3 \).

1. Identify \( a \) and \( b \) such that \( a^3 = s^3 \) and \( b^3 = 729 \).
2. Apply the difference of cubes formula.

The expression \( s^3 - 729 \) can be recognized as a difference of cubes, where \( a = s \) and \( b = 9 \) since \( 729 = 9^3 \).

Using the difference of cubes formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] we substitute \( a \) and \( b \): \[ s^3 - 9^3 = (s - 9)(s^2 + 9s + 81) \]

Final Answer