fraction decomposition - where is the mistake

[jozwoz99]

Hi

When I try to decompose: (x + 1) / ((x + 2)^2)

I end up with: x + 1 = A(x + 2)^2 + B(x + 2)

this ends up with A = 1/2, B = - 1/2, so the answer seemingly should be:

1/2(x + 2) - 1/2(x + 2)^2

This doesn't make sense however as the result when these terms are added back is

(x + 1) / 2(x + 2)^2 rather than (x + 1) / (x + 2)^2

Can someone explain the result to me\

Thanks

[jg.allinsymbols]

Given , decompose into denominations of this form:


Work the steps to perform the addition and then compare the numerator terms to solve for A and B.

.
.
Toward you last steps, you should obtain the two simultaneous equations, Ax=x and 2A+B=1. If not, show your work and someone may identify your mistake.

[nona.m.nona]

When I try to decompose: (x + 1) / ((x + 2)^2)

I end up with: x + 1 = A(x + 2)^2 + B(x + 2)

How? Kindly please reply with your steps. Thank you.

[jg.allinsymbols]

When I try to decompose: (x + 1) / ((x + 2)^2)

I end up with: x + 1 = A(x + 2)^2 + B(x + 2)

How? Kindly please reply with your steps. Thank you.

=



And grouping to begin seeing correspondence of the numerators,


Now, carefully equate the corresponding parts of the numerator:
and because you are comparing x+1 with Ax+2A+B
The coefficient on x is 1, so A=1.
Using this for A in the constants relationship, 2*1+B=1 means B=-1