Final answer:
In Part A, we add the two given polynomials: ab² + 3ab + 8a² and -5ab². In Part B, we multiply the two polynomials: ab² + 3ab + 8a² and -5ab². The question seems to have a duplicate in Part C.
Step-by-step explanation:
Part A: In order to demonstrate that polynomials are closed under addition, we need to add the two given polynomials: ab² + 3ab + 8a² and -5ab². Since both polynomials have the same variables (a and b), we can combine like terms. Combining the like terms, we get ab² + 3ab + 8a² + (-5ab²), which simplifies to 8a² + 3ab - 4ab².
Part B: To show that polynomials are closed under multiplication, we need to multiply the two given polynomials: ab² + 3ab + 8a² and -5ab². Multiplying the two polynomials, we get (ab² + 3ab + 8a²) * (-5ab²). Using the distributive property, we multiply each term of the first polynomial by each term of the second polynomial. Simplifying the expression, we get -5a³b⁴ - 15a²b³ - 40a³b².
Part C: It seems like Part C is a duplicate of Part B. Please provide the correct question or clarify Part C.