MQ61 --- Quartic Equations

Let F(n) = ax^n + by^n

Notice that (x + y)F(n) - xyF(n - 1)

=(x + y)(ax^n + by^n) - xy(ax^n-1 + by^n-1)

= ax^n+1 + bxy^n + ax^ny + by^n+1 - ax^ny - bxy^n

= ax^n+1 + by^n+1

= F(n + 1)

We have

F(3) = (x + y)F(2) - xyF(1) => 16 = 7(x + y) - 3xy

F(4) = (x + y)F(3) - xyF(2) => 42 = 16(x + y) - 7xy

(x + y) = -14, xy = -38

So, ax^5 + by^5

= F(5)

= (x + y)F(4) - xyF(3)

= (-14)(42) - (-38)(16)

= 20