Verify that f(x)= (1-x^2) - (2+x) can be written as: -x + 2 - 3/(2+x)

[suegee3000]

Homework Statement

verify that f(x)= (1-x^2) - (2+x) can be written as: -x + 2 - 3/(2+x)

Homework Equations

The Attempt at a Solution

i've tried manipulating as 1/(2+x) -x^2/(2+x), multiplying the orig. equation by (2-x)/(2-x), factoring, etc. nothing's worked, and i can't think of another approach. can anyone steer me in the right direction? thank you!

 

[berkeman]

Homework Statement

verify that f(x)= (1-x^2) - (2+x) can be written as: -x + 2 - 3/(2+x)

Homework Equations

The Attempt at a Solution

i've tried manipulating as 1/(2+x) -x^2/(2+x), multiplying the orig. equation by (2-x)/(2-x), factoring, etc. nothing's worked, and i can't think of another approach. can anyone steer me in the right direction? thank you!

So basically you want to show:

$$(1-x^2) - (2+x) = -x + 2 - \frac{3}{2+x}$$

I'd be inclined to put the RHS over a common denominator, and then look to see what the next steps would be to distribution things out and start looking for simplifications...

 

[HallsofIvy]

On obvious point should be that the right hand side is NOT defined for x= -2 while the left hand side is. They can't possibly be equal! At first I thought that you mean "show they are equal for all x except -2" but then I notice that if x= 0, the left hand side is 1- 2= -1 while the right hand side is 1/2. And if x= 1, the left side is -3 while the right side is 0. Do you see my point?