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Binomial Theorem

  1. Jun 2, 2012 #1
    The problem statement, all variables and given/known data
    The method of Binomial expansion is useful because you can avoid expanding large expressions:
    Q: Find the term indepedent of x in the expansion of (2+x)[2x+(1/x)]5

    The attempt at a solution:
    "For this to produce a term independent of x, the expansion of [2x+(1/x)]5 must have a constant term or a term in x-1...?
    So the power of x is given by 5-2r. This cannot be zero for positive integer values of r. Hence the required coefficient is given by:
    5 - 2r = -1
    r = 3"

    But why?! I do not understand that underlined statement.
     
  2. jcsd
  3. Jun 2, 2012 #2
    What would be happen if it becomes zero? See there's a term (2+x) also present. :wink:
     
  4. Jun 2, 2012 #3
    So...the expansion of [2x+(1/x)]5 must have a constant term or a term in x-1...

    Constant term means the term with no x right?

    But why can the term independent of x be in x-1 too?
     
  5. Jun 2, 2012 #4
    Because the term already has a x-1!! It cannot be independent of x if it already has a x :rolleyes:

    To explain the question, you need a term independent of x. You get them independent of x, if a constant term gets multiplied with another constant term, OR a term with x-1 is multiplied with a term with x. But there is no constant term(not involving x) in the expansion of [2x+(1/x)]5 as explained. So the only possibility remains, that x-1 gets multiplied with x.
     
  6. Jun 2, 2012 #5
    I am so lost here.. why am I so terrible at maths :frown:
     
  7. Jun 2, 2012 #6
    Yes! :smile:

    We need to find the constant term in expansion of (2+x)(2x+(1/x))5, not in (2x+(1/x))5. So when x in (2+x) multiplies by a term having x-1 with it, we get a constant term as [itex]x*\frac{1}{x}=1.[/itex].
     
  8. Jun 2, 2012 #7
    Ahh.. I think I'm starting get the grasp of this.. one more problem!
    Expand (2+x)5 and hence find 1.95
    = 32 + 80x+ 80x2 + 40x3 + 10x4 + x5

    Why can 1.95 be considered to be x= -0.1 in the above expansion?
     
  9. Jun 2, 2012 #8
    What is (2+(-0.1))5=?
     
  10. Jun 2, 2012 #9
    Thanks very much. Can't believe I can't even think simple.
     
  11. Jun 5, 2012 #10
    its binomial theorem.
    use the formula t(r+1)=nCr*x^n-r*a6r
    x=x^2 and a=-1/2x (substitution of the question variables)
    n=power=3
    let the term independent of x be t(r+1)=x^0=1
    therefore after substitution the formula would be
    Y*x^0=nCr*(x^2)^3-r*(-1*x^ -1)^r
    therefore power of x=6-2r-r
    but in the term power of x is supposed to be 0
    0=6-3r
    6=3r
    r=2.
    therefore the term is r+1
    2+1
    =3..
    therefore the term independent of x is the third term.
     
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