Which is a difference of cubes?
x6 – 27
x15 – 36
x16 – 64
x5 – 125
The Correct Answer and Explanation is :
A difference of cubes is an algebraic expression that can be represented in the form ( a^3 – b^3 ), which can be factored using the formula:
[
a^3 – b^3 = (a – b)(a^2 + ab + b^2)
]
To determine which of the expressions provided is a difference of cubes, we can analyze each one:
- ( x^6 – 27 ):
- This can be rewritten as ( (x^2)^3 – 3^3 ).
- Here, ( a = x^2 ) and ( b = 3 ), so it is a difference of cubes.
- ( x^{15} – 36 ):
- This does not fit the form ( a^3 – b^3 ) because ( 36 ) is not a perfect cube.
- The cube root of ( 36 ) is not an integer.
- ( x^{16} – 64 ):
- This can be rewritten as ( (x^{16/3})^3 – 4^3 ).
- However, ( x^{16/3} ) is not a simple term like ( x^n ), so this is not a standard difference of cubes.
- ( x^5 – 125 ):
- This can be rewritten as ( x^5 – (5)^3 ).
- Here, ( a = x^{5/3} ) and ( b = 5 ), which again complicates it.
Conclusion
The only expression among the four provided that clearly fits the difference of cubes format is ( x^6 – 27 ). This expression can be factored as:
[
x^6 – 27 = (x^2 – 3)(x^4 + 3x^2 + 9)
]
Explanation
In algebra, recognizing patterns such as the difference of cubes is essential for simplifying expressions and solving equations. The structure ( a^3 – b^3 ) represents the difference between two cubes, where each component can be expressed as a cubic term. The factorization not only provides a simpler form but also reveals the roots of the equation when set to zero. Understanding how to manipulate these forms is a fundamental skill in algebra that has applications across calculus, physics, engineering, and beyond.