Question

what is the factored form of (27d^6 + 8g^{12})?

• ((3d^2 + 2g^4)(3d^2 - 6d^2g^4 + 2g^4))
• ((3d^2 + 2g^4)(9d^4 - 6d^2g^4 + 4g^8))
• ((3d^2 + 2g^4)(9d^4 + 6d^2g^4 + 4g^8))
• ((3d^2 + 2g^4)(3d^4 - 6d^2g^4 + 2g^8))

Explanation:

Identify the terms as perfect cubes

$$ 27d^6 + 8g^{12} = (3d^2)^3 + (2g^4)^3 $$

Apply the sum of cubes formula

$$ A^3 + B^3 = (A + B)(A^2 - AB + B^2) $$

$$ A = 3d^2,\quad B = 2g^4 $$

Expand the factored terms

$$
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$$

$$ (3d^2 + 2g^4)(9d^4 - 6d^2g^4 + 4g^8) $$

Answer:

• (A) \((3d^2 + 2g^4)(3d^2 - 6d^2g^4 + 2g^4)\)
• (B) \((3d^2 + 2g^4)(9d^4 - 6d^2g^4 + 4g^8)\) (Correct answer)
• (C) \((3d^2 + 2g^4)(9d^4 + 6d^2g^4 + 4g^8)\)
• (D) \((3d^2 + 2g^4)(3d^4 - 6d^2g^4 + 2g^8)\)