Question

(a-b)(a+2b)^2-(a+b)^3+b^3 разложить на множители

Answer 1

Вот так. Надеюсь поймешь мой ужасный почерк

Answer 2

To factorize the given expression "(a-b)(a+2b)^2-(a+b)^3+b^3," we'll start by simplifying it step by step:

Step 1: Expand both squares (a+2b)^2 = (a+2b)(a+2b) = a^2 + 4ab + 4b^2

Step 2: Expand the cubed term (a+b)^3 = (a+b)(a+b)(a+b) = (a+b)(a^2 + 2ab + b^2) = a^3 + 3a^2b + 3ab^2 + b^3

Now, we can rewrite the expression using the expanded forms:

(a-b)(a+2b)^2 - (a+b)^3 + b^3 = (a-b)(a^2 + 4ab + 4b^2) - (a^3 + 3a^2b + 3ab^2 + b^3) + b^3

Step 3: Distribute the (a-b) term = a(a^2 + 4ab + 4b^2) - b(a^2 + 4ab + 4b^2) - (a^3 + 3a^2b + 3ab^2 + b^3) + b^3

Step 4: Combine like terms = a^3 + 4a^2b + 4ab^2 - a^2b - 4ab^2 - 4b^3 - a^3 - 3a^2b - 3ab^2 - b^3 + b^3

Step 5: Simplify further = a^3 - a^3 + 4a^2b - a^2b + 4ab^2 - 3ab^2 - 4b^3 + b^3

Step 6: Combine like terms again = 3a^2b + ab^2 - 4b^3

Now, the expression is simplified, and it's not possible to factorize it further as the resulting expression doesn't have any common factors or recognizable patterns.