Question

Which of the following is a factor of (x+y)³ - (x³+y³) ?

choose the correct answer
x²+xy²+2xy
x²+y²-xy
xy²
3xy

Answer 1

3 x y (y + x)

Explanation:

Factor the following:

(y + x)^3 - (y^3 + x^3)

Factor the sum of two cubes. y^3 + x^3 = (y + x) (y^2 - x y + x^2):

(y + x)^3 - ((y + x) (y^2 - x y + x^2))

Factor y + x out of (y + x)^3 - (y + x) (y^2 - x y + x^2), resulting in (y + x) ((y + x)^(3 - 1) - (y^2 - x y + x^2)):

(y + x) ((y + x)^(3 - 1) - (y^2 - x y + x^2))

3 - 1 = 2:

(y + x) ((y + x)^2 - (y^2 - x y + x^2))

(y + x) (y + x) = (x) (x) + (x) (y) + (y) (x) + (y) (y) = x^2 + x y + x y + y^2 = y^2 + 2 x y + x^2:

(y + x) ((y^2 + 2 x y + x^2) - (y^2 - x y + x^2))

-(y^2 - x y + x^2) = -y^2 + x y - x^2:

(y + x) (x^2 + 2 x y + y^2 + (-y^2 + x y - x^2))

Grouping like terms, y^2 - y^2 + 2 x y + x y + x^2 - x^2 = (2 x y + x y) + (y^2 - y^2) + (x^2 - x^2):

((2 x y + x y) + (y^2 - y^2) + (x^2 - x^2)) (y + x)

x y 2 + x y = x y×3:

(3 x y + (y^2 - y^2) + (x^2 - x^2)) (y + x)

y^2 - y^2 = 0:

(3 x y + (x^2 - x^2)) (y + x)

x^2 - x^2 = 0:

Answer: 3 x y (y + x)