Question

A)(a+b)(a² −ab+b²)

b) (a−b)(a² +ab+b²) Questions:

A) What are the resulting products?
B) How are the terms of the products related to the terms of the factors?
C) What if the process was reversed and you were asked to find the factors of the products?
D) How are you going to get the factor? Do you see any common patterns?

Answer 1

In the expression (a+b)(a² −ab+b²), the resulting product is a³+b³. In the expression (a−b)(a² +ab+b²), the resulting product is a³−b³. The common patterns observed are the sums or differences of cubes.

Step-by-step explanation:

In the expression (a+b)(a² −ab+b²), the resulting product is:

(a+b)(a² −ab+b²) = a(a² −ab+b²) + b(a² −ab+b²) = a³ −a²b+ab² + a²b −ab²+b³ = a³+b³

In the expression (a−b)(a² +ab+b²), the resulting product is:

(a−b)(a² +ab+b²) = a(a² +ab+b²) − b(a² +ab+b²) = a³ +a²b+ab² − a²b−ab²−b³ = a³−b³

In both cases, the resulting products are of the form a³+b³ or a³−b³.

If the process is reversed and we are asked to find the factors of the products, we can use the binomial theorem to expand the products and identify the factors.

The common patterns we observe in the resulting products are the sums or differences of cubes, which can be expressed as a³+b³ or a³−b³.