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# The integers G and H are chosen

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The integers G and H are chosen such that

G/(x + 5) + H/(x^2 - 4x) = (3x^2 - 13x - 5)/(x^3 + x^2 - 20x)

for all values of x except -5, 0, and 4.  Find H/G.

Nov 19, 2020

#1
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See many, many integer solutions here:

https://www.wolframalpha.com/input/?i=y%2F%28x%2B5%29%2Bz%2F%28x%5E2-4*x%29%3D%283*x%5E2-13*x-5%29%2F%28x%5E3%2Bx%5E2-20*x%29%2C+integer+solution

Nov 19, 2020
#2
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G/(x + 5) + H/(x^2 - 4x) = (3x^2 - 13x - 5)/(x^3 + x^2 - 20x)

$$\frac{G}{(x + 5) }+ \frac{H}{(x^2 - 4x)} = \frac{(3x^2 - 13x - 5)}{(x^3 + x^2 - 20x)}\\ \frac{G}{(x + 5) }+ \frac{H}{x(x - 4)} = \frac{(3x^2 - 13x - 5)}{x(x+5)(x-4)}\\ \frac{G*x(x - 4)+H(x+5)}{{x(x+5)(x-4)} }= \frac{(3x^2 - 13x - 5)}{x(x+5)(x-4)}\\ G*x(x - 4)+H(x+5)=3x^2 - 13x - 5$$

Expand then simplify the LHS

Then equate coefficients

Nov 19, 2020