7. \( 27 a^{3}-1 \)

Did you know that \( 27a^3 - 1 \) is a special case known as a difference of cubes? You can factor it using the formula \( x^3 - y^3 = (x - y)(x^2 + xy + y^2) \). Here, \( x = 3a \) and \( y = 1 \), which gives us \( (3a - 1)(9a^2 + 3a + 1) \). It's a neat mathematical trick that simplifies calculations and helps in graphing functions! To solve problems involving expressions like \( 27a^3 - 1 \), be cautious about common mistakes. A frequent error is attempting to factor it as a simple binomial without recognizing it's a difference of cubes. Also, be mindful of the signs in your factorization; they can lead to the wrong answer if overlooked. Always double-check your work by expanding your factors to ensure you return to the original expression!