+0  
 
0
40
1
avatar+117 

The integers \(G\) and\(H\) are chosen such that \(\frac{G}{x+5}+\frac{H}{x^2-4x}=\frac{x^2-2x+10}{x^3+x^2-20x}\) for all real values of \(x\) except -5, 0, and 4. Find \(H/G\).

DanielCai  Jun 29, 2018
 #1
avatar+87294 
+2

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

 

 

 

cool cool cool

CPhill  Jun 29, 2018

8 Online Users

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.