Factor completely.

64
𝑥
3
−
𝑦
3

Question

Factor completely.

64
𝑥
3
−
𝑦
3

BlackTom AI · Accepted Answer

Let’s examine the expression to factor: 64x^3 − y^3. This is a difference of cubes form, since 64x^3 is (4x)^3 and y^3 is y^3.
Option A: Prime — This is not prime factoring; 64x^3 − y^3 can be decomposed into binomial and trinomial factors, so labeling it prime misses the standard algebraic identity for a difference of cubes.

Option B: (4x − y)(4x^2 + x y + y^2) — Compare with the standard formula a^3 − b^3 = (a − b)(a^2 + a b + b^2). If we set a = 4x and b = y, then a^3 − b^3 = (4x − y)(16x^2 + 4x y + y^2). In this option, the quadratic factor is missing a factor of 4 in the x^2 term, so the coefficient pattern is incorrect. Therefore this is not the correct factorization.

Option C: (4x + y)(4x^2 − x y + y^2) — This matches a sum of cubes pattern flipped, which would apply to (4x)^3 + y^3, not to (4x)^3 − y^3. Since the original is a difference, this option is inappropriate.

Option D: (4x − y)(16x^2 + 4xy + y^2) — This aligns precisely with the a^3 − b^3 identity using a = 4x and b = y. Expanding would yield (4x)^3 − y^3 = 64x^3 − y^3, because (4x − y)(16x^2 + 4xy + y^2) = 64x^3 − y^3. This is the standard and correct factorization for the given expression.

Option E: (4x + y)(16x^2 − 4xy + y^2) — This structure corresponds to a^3 + b^3 factorization with the plus sign inside the first factor but a minus sign in the middle term of the quadratic; it does not produce a difference of cubes when expanded. Therefore it’s incorrect here.

Option F: (4x^2 − y^2)(16x + y) — This resembles a difference of squares and a linear factor, but it does not reflect the a^3 − b^3 pattern. Expanding would not reproduce 64x^3 − y^3, so this is incorrect.

Option G: (4x^2 + y^2)(16x − y) — This pairing doesn’t correspond to any standard a^3 − b^3 decomposition and fails to expand to 64x^3 − y^3. Thus it’s not a valid factorization here.

In summary, the only option that matches the difference-of-cubes factoring pattern with a = 4x and b = y is the one that yields (4x − y)(16x^2 + 4xy + y^2) upon expansion. The other choices either misplace signs, drop necessary coefficients, or do not align with the a^3 − b^3 identity, leading to incorrect expansions.

H Algebra 2 HA2 4.5/4.6 Quiz

Factor completely. 64 𝑥 3 − 𝑦 3

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