Question

What is the factored form of 27d^6+8g^12?

a) (3d^2 + 2g^4)(9 - 6g^8 + 20g^4)
b) [3d + 2g]^6(9 - 6g^6 + 4g^2)
c) (3d^2 + 2g^4)(9 - 6g^6 + 4g^2)
d) [3d + 2g]^6(9 - 6g^8 + 4g^4)

Answer 1

The factored form of 27d^6+8g^12 is (3d^2 + 2g^4)(9d^4 - 6d^2g^4 + 4g^8), which is a sum of cubes and corresponds to option (c).

Step-by-step explanation:

The factored form of 27d^6+8g^12 is obtained by recognizing it as a sum of two cubes, since 27d^6 = (3d^2)^3 and 8g^12 = (2g^4)^3. Applying the sum of cubes formula, a^3 + b^3 = (a+b)(a^2 - ab + b^2), we get:

1. Identify a = 3d^2 and b = 2g^4.
2. Apply the formula: (3d^2+2g^4)((3d^2)^2 - (3d^2)(2g^4) + (2g^4)^2).
3. Simplify the expression: (3d^2+2g^4)(9d^4 - 6d^2g^4 + 4g^8).

Therefore, the correct answer is (3d^2 + 2g^4)(9d^4 - 6d^2g^4 + 4g^8), which corresponds to option (c).