What is the complete factorization of 8x3 - 125?

Question

BlackTom AI · Accepted Answer

To factor 8x^3 - 125, recognize it as a difference of cubes: (2x)^3 - 5^3. The standard factorization for a^3 - b^3 is (a - b)(a^2 + ab + b^2).

Option a: (2x + 5)(4x^2 - 10x + 25)
- This switches the sign of the linear term inside the second factor and also changes the sign of the first binomial, which does not align with the a^2 + ab + b^2 pattern. If you expand, you would not recover 8x^3 - 125, so this is incorrect.

Option b: (2x - 5)^3
- This would expand to 8x^3 - 60x^2 + 150x - 125, a completely different polynomial with x^2 and x terms present. Since the original expression has only x^3 and constant terms, this option cannot be correct.

Option c: (2x - 5)(4x^2 - 10x + 25)
- Here, the second factor is 4x^2 - 10x + 25, which includes a negative 10x term. In the a^2 + ab + b^2 form, the middle term should be + ab; ab equals 10x, not -10x. This sign mismatch means this factorization would not yield 8x^3 - 125 when expanded, so it is incorrect.

Option d: (2x - 5)(4x^2 + 10x + 25)
- This matches the correct pattern: a = 2x, b = 5, with a - b = 2x - 5 and a^2 + ab + b^2 = 4x^2 + (2x)(5) + 25 = 4x^2 + 10x + 25. Expanding gives (2x)(4x^2 + 10x + 25) - 5(4x^2 + 10x + 25) = 8x^3 + 20x^2 + 50x - 20x^2 - 50x - 125 = 8x^3 - 125, matching the original expression.

In summary, each incorrect option fails due to sign errors, incorrect exponent patterns, or expansion results that do not reproduce the original polynomial. The option that aligns with the difference-of-cubes formula and yields the original expression upon expansion is the one with (2x - 5)(4x^2 + 10x + 25).

What is the complete factorization of 8x3 - 125?单项选择题

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