QuestionDecember 17, 2025

38. Use polynomial identities to factor. -27x^3+125 Type a response

38. Use polynomial identities to factor. -27x^3+125 Type a response
38. Use polynomial identities to factor.
-27x^3+125
Type a response

Solution
4.4(281 votes)

Answer

-(3x - 5)(9x^2 + 15x + 25) Explanation 1. Recognize the sum/difference of cubes -27x^3 + 125 is a difference of cubes: -(27x^3 - 125). 2. Apply the difference of cubes formula Use a^3 - b^3 = (a-b)(a^2 + ab + b^2) with a = 3x, b = 5. 3. Factor using the formula -(27x^3 - 125) = -(3x - 5)\left((3x)^2 + (3x)(5) + 5^2\right). 4. Simplify the factors (3x)^2 = 9x^2, (3x)(5) = 15x, 5^2 = 25.

Explanation

1. Recognize the sum/difference of cubes<br /> $-27x^3 + 125$ is a difference of cubes: $-(27x^3 - 125)$.<br />2. Apply the difference of cubes formula<br /> Use $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ with $a = 3x$, $b = 5$.<br />3. Factor using the formula<br /> $-(27x^3 - 125) = -(3x - 5)\left((3x)^2 + (3x)(5) + 5^2\right)$.<br />4. Simplify the factors<br /> $(3x)^2 = 9x^2$, $(3x)(5) = 15x$, $5^2 = 25$.
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