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A Factor Theorem Question

  1. Mar 25, 2003 #1
    My maths teacher says this problem is not as impossible as it seems, but I just can't solve it.

    Show that (x + a + b)^7 - x^7 - a^7 - b^7 is divisble by
    x^2 + (a + b)x +ab.
     
  2. jcsd
  3. Mar 25, 2003 #2

    drag

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    Science Advisor

    Looking for the easy way out huh ?
    You can always solve the entire excercise...

    Live long and prosper.
     
  4. Mar 26, 2003 #3
    Hint:
    Notice that (x+a) and (x+b) are the 2 factors of x^2 + (a + b)x +ab.
    So it is equivalent to show that (x + a + b)^7 - x^7 - a^7 - b^7 is divisible by both (x+a) and (x+b).

    Let f(x) = (x + a + b)^7 - x^7 - a^7 - b^7
    .....
    .....
    .....
    .....


    Can you continue from here?

    Hope this help. :smile:
     
  5. Mar 26, 2003 #4
    just write everything out
    eg. (x+a)^2=x^2+2xa+a^2

    maybe rewrite some terms then and you will see that it is divisible by x^2 + (a + b)x +ab
     
  6. Mar 26, 2003 #5

    Hurkyl

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    Gold Member

    KL has the easy way!

    Writing it out however... *shudder* I wouldn't wish writing out a trinomial to the 7th power to anyone!

    Hurkyl
     
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