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Finding the remainder of an algebraic quotient

  1. Dec 7, 2014 #1
    I'm tutoring a pupil for a CLEP exam and her book includes the following algebra problem:
    What is the remainder when
    9x^{23} - 7x^{12} - 2x^{5} +1
    is divided by [itex] x+1 [/itex]?
    I know how to find the answer by computing the quotient of these two expressions, but in this case doing that is so tedious I assume there's a more direct way of finding the remainder. What is it?

    Edit : I think this might be more appropriately placed in the "Homework and Coursework" section.
    Last edited: Dec 7, 2014
  2. jcsd
  3. Dec 7, 2014 #2
    Ruffini's rule speeds up the proces when divisior is in form "x-r"
  4. Dec 7, 2014 #3
    Indeed it does, thanks zoki85!
  5. Dec 8, 2014 #4


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    Since the problem is only to find the remainder, even simpler is "The remainder when dividing polynomial function P(x) by x- r is P(r)".
    That's easy to prove: let Q(x) be the quotient when P(x) is divided by x- r. The P(x)= Q(x)(x- r)+ remainder. Letting x= r give P(r)= Q(r)(0)+ remainder or "remainder= P(r)". To find the remainder when [itex]P(x)= 9x^{23}- 7x^{12}- 2x^5+ 1[/itex] is divided by x+1= x- (-1), just calculate
    [tex]P(-1)= 9(-1)^{23}- 7(-1)^{12}- 2(-1)^5+ 1[/tex].
    Last edited by a moderator: Dec 8, 2014
  6. Dec 8, 2014 #5
    Wow! (high fives HallsofIvy)
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