+145 The integers G andH are chosen such that Gx+5+Hx2−4x=x2−2x+10x3+x2−20x for all real values of x except -5, 0, and 4. Find H/G.
+130474 We can use partial fractions, here
Note that x^3 + x^2 -20x can be factored as ( x^2 - 4x) ( x + 5)
x^2 - 2x + 10 G H
______________ = _____ + ________
( x^2 - 4x)(x + 5) x + 5 x^2 - 4x
Multiply through by ( x + 5) ( x^2 - 4x)
x^2 - 2x + 10 = G(x^2 - 4x) + H (x + 5) simplify
x^2 - 2x + 10 = Gx^2 - 4Gx + Hx + 5H equate coefficients
1 = G
-2 = H - 4G
10 = 5H
It's obvious that G = 1 and H = 2
So
H / G = 2 / 1 = 2
