Question

Tomas learned that the product of the polynomials (a + b)(a2 – ab + b2) was a special pattern that would result in a sum of cubes, a3 + b3. His teacher put four products on the board and asked the class to identify which product would result in a sum of cubes if a = 2x and b = y.

Which product should Tomas choose?

(2x + y)(2x2 + 2xy – y2)
(2x + y)(4x2 + 2xy – y2)
(2x + y)(4x2 – 2xy + y2)
(2x + y)(2x2 – 2xy + y2)

Answer 1

Use the identity (a + b)(a^2 − ab + b^2) = a^3 + b^3. With a = 2x and b = y, a^2 − ab + b^2 = 4x^2 − 2xy + y^2.

So Tomas should choose the third product: (2x + y)(4x^2 − 2xy + y^2), which equals (2x)^3 + y^3 = 8x^3 + y^3.