factoring a sum of cubes

factor $125x^9 + 64$.

$(5x^3 - 4)(25x^6 + 20x^3 + 16)$
$(5x^3 - 4)(25x^3 + 20x^3 + 16)$
$(5x^3 + 4)(25x^6 - 20x^3 + 16)$
$(5x^3 + 4)(25x^3 - 20x^3 + 16)$

Question

factoring a sum of cubes

factor $125x^9 + 64$.

$(5x^3 - 4)(25x^6 + 20x^3 + 16)$
$(5x^3 - 4)(25x^3 + 20x^3 + 16)$
$(5x^3 + 4)(25x^6 - 20x^3 + 16)$
$(5x^3 + 4)(25x^3 - 20x^3 + 16)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Sum of Cubes" ], "new_concepts": [], "current_concepts": [ "Sum of Cubes", "Factoring Polynomials" ] } ## Rewrite terms as perfect cubes Using the *Sum of Cubes* knowledge point $$ \begin{aligned} 125x^9 &= (5x^3)^3 \ 64 &= 4^3 \end{aligned} $$ ## Apply the sum of cubes formula Using the *Sum of Cubes* knowledge point $$ \begin{aligned} a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \ a &= 5x^3, \quad b = 4 \ 125x^9 + 64 &= (5x^3 + 4)((5x^3)^2 - (5x^3)(4) + 4^2) \end{aligned} $$ ## Simplify the factored expression Using the *Sum of Cubes* knowledge point $$ \begin{aligned} (5x^3)^2 &= 25x^6 \ (5x^3)(4) &= 20x^3 \ 4^2 &= 16 \ 125x^9 + 64 &= (5x^3 + 4)(25x^6 - 20x^3 + 16) \end{aligned} $$

(A) $(5x^3 - 4)(25x^6 + 20x^3 + 16)$

(B) $(5x^3 - 4)(25x^3 + 20x^3 + 16)$

(C) $(5x^3 + 4)(25x^6 - 20x^3 + 16)$

(D) $(5x^3 + 4)(25x^3 - 20x^3 + 16)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Sum of Cubes" ] }

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QUESTION IMAGE

Question

Explanation:

Rewrite terms as perfect cubes

Apply the sum of cubes formula

Simplify the factored expression

Answer: