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How would I express (2x - 1) / (x + 2) as A + B / (x+2), where A and B are integers?

I can do 2x / (x + 2) - 1 / (x+2) of course, but that doesn't work...

Any ideas?

 

EDIT: Sorry if I wasn't clear, but it's NOT:

 $${\frac{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\right)}} = {\mathtt{A}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{B}}}{\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\right)}}$$

But instead I need to change:

$${\frac{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\right)}}$$

Into something that looks like:

$${\mathtt{A}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{B}}}{\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\right)}}$$

Thank you!

Will85237  Jan 19, 2015

Best Answer 

 #2
avatar+91412 
+10

I think anon's logic should work but I would do it much more simply.

 

$$\frac{2x-1 }{ x+2}\\\\
=\frac{2(x+2)-1-4}{x+2}\\\\
=\frac{2(x+2)-5}{x+2}\\\\
=\frac{2(x+2)}{x+2}+\frac{-5}{x+2}\\\\
=2+\frac{-5}{x+2}$$

 

Don't give anon negative points for trying to help you!!!

Melody  Jan 19, 2015
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3+0 Answers

 #1
avatar
+5

Start by multiplying both sides by (x+2), giving 2x-1 = A(x+2)+B

Now, multiply out and equate x terms on the left to x terms on the right, and we have A=2, and also -1=2A+B

Now just solve for B. 🚲

Guest Jan 19, 2015
 #2
avatar+91412 
+10
Best Answer

I think anon's logic should work but I would do it much more simply.

 

$$\frac{2x-1 }{ x+2}\\\\
=\frac{2(x+2)-1-4}{x+2}\\\\
=\frac{2(x+2)-5}{x+2}\\\\
=\frac{2(x+2)}{x+2}+\frac{-5}{x+2}\\\\
=2+\frac{-5}{x+2}$$

 

Don't give anon negative points for trying to help you!!!

Melody  Jan 19, 2015
 #3
avatar+91412 
+5

YES anon understood what you wanted and gave you a correct answer!

I shall show you

 

$$\\\frac{2x-1}{x+2}=A+\frac{B}{x+2}\\\\
$multiply both sides by (x+2)$\\
2x-1=A(x+2)+B\\
2x-1=Ax+2A+B\\
2x-1=Ax+(2A+B)\\
$ equating co-efficients$\\
2=A\\
-1=2A+B\\
-1=4+B\\
B=-5\\
so\\
\frac{2x-1}{x+2}=2+\frac{-5}{x+2}\\\\$$

 

so maybe you owe anon an appology  

Melody  Jan 19, 2015

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