Question

Find the fourth proportional to:
x³ - y², x⁴ + x²y² + y⁴, x - y.

Answer 1

Let A be the fourth proportional then
x³ - y² : x⁴ + x²y² + y⁴ = x - y : A
⇒ `(x^3 - y^3)/(x^4 + x^2y^2 + y^4) = (x - y)/"A"`
⇒ A(x³ - y³) = (x - y)(x⁴ + x²y² + y⁴)
⇒ A = `((x - y)(x^4 + x^2y^2 + y^4))/(x^3 - y^3)`
⇒ A = `((x - y)(x^2 + y^2 + xy)(x^2 + y^2 - xy))/((x - y)(x^2 + xy + y^2)`
⇒ A = x² + y² - xy.