Final Answer:
The correct factorization is (B) (5x⁵y⁶ + 6z³)(7x⁵y⁶ - 6z³).
Step-by-step explanation:
To factorize the given expression (35x¹⁰y¹² - 69x⁸y¹⁰z⁸ - 36z¹³, we can observe that the common factor among the three terms is (z³). Factoring out (z³) from each term, we get z³(35x¹⁰y¹² - 69x⁸y¹⁰z⁵ - 36z⁸).
Now, we can further factorize the quadratic expression within the parentheses. It resembles the form (a² - b²), which factors into (a + b)(a - b). In this case, let (a = 5x⁵y⁶) and (b = 6z³). Applying the difference of squares formula, we get z³(5x⁵y⁶ + 6z³)(7x⁵y⁶ - 6z³).
Therefore, the correct factorization is z³(5x⁵y⁶ + 6z³)(7x⁵y⁶ - 6z³), which matches option (B) z³(5x⁵y⁶ + 6z³)(7x⁵y⁶ - 6z³). This choice correctly represents the factorized form of the given expression.
In conclusion, through the process of factoring out the common factor and applying the difference of squares formula, we arrive at the factorization represented by option (B).