Which expression gives the factorization of $x^3-27 y^3$ ?

A. $(x-3 y)(x^2+3 x y+9 y^2)$
B. $(x-3 y)(x^2-3 x y+3 y^2)$
C. $(x-3 y)(x^2+3 x y+3 y^2)$
D. $(x-3 y)(x^2-3 x y+9 y^2)$

Recognize the expression as a difference of cubes: x 3 − ( 3 y ) 3 .
Apply the difference of cubes factorization formula: a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) .
Substitute a = x and b = 3 y into the formula: ( x − 3 y ) ( x 2 + x ( 3 y ) + ( 3 y ) 2 ) .
Simplify the expression: ( x − 3 y ) ( x 2 + 3 x y + 9 y 2 ) .
The correct factorization is ( x − 3 y ) ( x 2 + 3 x y + 9 y 2 ) ​ .

Explanation

Recognizing the Problem Type We are asked to factorize the expression x 3 − 27 y 3 . This expression is a difference of cubes.

Recalling the Difference of Cubes Formula The difference of cubes factorization formula is given by: a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) .

Applying the Formula In our case, we have x 3 − 27 y 3 = x 3 − ( 3 y ) 3 . So, we can identify a = x and b = 3 y . Substituting these into the formula, we get: x 3 − ( 3 y ) 3 = ( x − 3 y ) ( x 2 + x ( 3 y ) + ( 3 y ) 2 ) .

Simplifying the Expression Now, we simplify the expression: ( x − 3 y ) ( x 2 + 3 x y + 9 y 2 ) .

Identifying the Correct Option Comparing this result with the given options, we find that the correct factorization is (x-3 y)\(x^2+3 x y+9 y^2) .


Examples
Factoring the difference of cubes is useful in simplifying algebraic expressions and solving equations. For example, if you have a volume calculation involving the difference of two cubes, factoring can help you find dimensions or simplify the expression for further analysis. Imagine you are designing a mold for a plastic part, and the part's volume is expressed as x 3 − 8 . By recognizing this as x 3 − 2 3 , you can factor it into ( x − 2 ) ( x 2 + 2 x + 4 ) , which might help you optimize the mold design or calculate material requirements more efficiently.

Answered by GinnyAnswer | 2025-07-07

The expression x 3 − 27 y 3 can be factored as ( x − 3 y ) ( x 2 + 3 x y + 9 y 2 ) using the difference of cubes formula. Therefore, the correct option is (A).
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Answered by Anonymous | 2025-07-14