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How do you go from (3w-1)/(w+2) and get: 3-(7/(w+2))?

  1. Jun 20, 2010 #1
    How do you go from

    (3w-1)/(w+2)

    and get:

    3-(7/(w+2))?
     
  2. jcsd
  3. Jun 20, 2010 #2

    Mark44

    Staff: Mentor

    Re: Rearranging?

    Add 0 to the numerator like this: 3w - 1 = 3w + 6 - 6 - 1 = 3(w + 2) - 7
    Now split the rational expression into two parts.
     
  4. Jun 20, 2010 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: Rearranging?

    Another way to do the same thing: divide 3w- 1 by w+ 2. w, alone, divides into 3w 3 times, of course. (3w- 1)/(w+ 2)= 3+ ?. Multiplying both sides by w+ 2, 3w- 1= 3w+ 6- ?(w+2). ?(w+ 2)= (3w+ 6)- (3w- 1)= 7 so that ?= 7/(w+ 2).

    (3w- 1)/(w+ 2)= 3+ 7/(w+ 2).
     
  5. Jun 20, 2010 #4
    Re: Rearranging?

    What about this one?

    2x/((x+3)(x+1))

    Simplified to

    3/(x+3) - 1/(x+1)

    This is probably one of the harder parts of algebra for me for some reason. I dont understand the steps.
     
  6. Jun 20, 2010 #5

    Mark44

    Staff: Mentor

    Re: Rearranging?

    This is an example of rational function decomposition. In this case you need to rewrite the original expression as the sum of two simpler rational expressions.

    [tex]\frac{2x}{(x + 3)(x + 1)} = \frac{A}{x + 3} + \frac{B}{x + 1}[/tex]

    The equation above needs to be identically true -- true for all reasonable values of x (the ones for which the denominators aren't zero).

    Multiply both sides of the equation above by (x + 3)(x + 1) and solve for A and B.
     
  7. Jun 20, 2010 #6
    Re: Rearranging?

    This one is a bit trickier

    1/[(u^2)(u-1)(u+1)]


    to get


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



    I tried doing it they way you just showed me but I get stuck.
     
  8. Jun 20, 2010 #7

    Mark44

    Staff: Mentor

    Re: Rearranging?

    Here you have a repeated factor, so the decomposition has to look like this:
    [tex]\frac{1}{u^2(u - 1)(u + 1)} = \frac{A}{u} + \frac{B}{u^2} + \frac{C}{u - 1} + \frac{D}{u + 1} [/tex]

    As before, multiply both sides by u2(u - 1)(u + 1) and solve for A, B, C, and D.

    There should be some examples of decomposition in your book...
     
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