Question

Factorize: (a+b)^3 + 8c^3

A. (a+b+2c)((a+b)^2-2ab+4c^2)
B. (a+b+2c)(a^2+b^2+4c^2)
C. (a+b+2c)(a^2+b^2-2ab+4c^2)
D. (a+b+2c)(a^2+b^2+2ab+4c^2)

Answer 1

To factorize (a+b)³ + 8c³, use the sum of cubes formula A³ + B³ = (A + B)(A² - AB + B²), setting A = (a+b) and B = 2c. After substituting and simplifying, we obtain (a+b+2c)(a² + 2ab + b² + 4c² - 2ac - 2bc), which does not match the provided options.

Step-by-step explanation:

The question asks to factorize the expression (a+b)³ + 8c³. This is a sum of cubes since it can be written as A³ + B³, where A is (a+b) and B is 2c. A sum of cubes can be factorized using the formula A³ + B³ = (A + B)(A² - AB + B²).

Let's apply this formula to our expression:

First, identify A and B from the expression: A = (a+b) and B = 2c.Then, substitute A and B into the formula to get the factors: ((a+b) + 2c)((a+b)² - (a+b)(2c) + (2c)²).Finally, expand and simplify the second factor: (a+b+2c)(a² + 2ab + b² - 2ac - 2bc + 4c²).Combine like terms in the second factor to get the final answer: (a+b+2c)(a² + 2ab + b² - 2ac - 2bc + 4c²) = (a+b+2c)(a² + 2ab + b² + 4c² - 2ac - 2bc).

However, none of the given options matches exactly our simplified form, which suggests there might be a typo in the options or in the original expression.