+0  
 
-1
137
1
avatar+126 

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 \(H/G\).

 Jun 29, 2018
 #1
avatar+94545 
+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

 Jun 29, 2018

23 Online Users

avatar
avatar
avatar
avatar

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.