1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question on remainder theorem

  1. Apr 9, 2013 #1
    1. The problem statement, all variables and given/known data
    If a , b, c are distinct and p(x) is a polynomial in x which leaves remainders a,b,c on division by (x-a),(x-b),(x-c) respectively. Then the remainder on division of p(x) by(x-a)(x-b)(x-c) is

    2. Relevant equations
    As it is given that p(x) gives remainder a when divided by (x-a), so p(a) should be equal to a by remainder theorem.Similarl p(b) = b and p(c)=c.



    3. The attempt at a solution
    As (x-a)(x-b)(x-c) is a cubic polynomial, remainder can be max quadratic so I assume it to be px^2 + qx + r.Again by remainder theorem we will get 3 equations for p,q,r by using a,b,c. As we see that p(a) = a, p(b)=b,p(c)=c ; Then we can say that a,b,c will be roots of px^2 +x(q-1) + r. But a quadratic polynomial can have max 2 roots. Can u please tell me what did I do wrong here?
     
    Last edited: Apr 9, 2013
  2. jcsd
  3. Apr 9, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    How did you come to that conclusion?
     
  4. Apr 9, 2013 #3
    Because (x-a)(x-b)(x-c) will be 0 at the values ie a,b,c so remainder at each of these values will be p(a),p(b),p(c) ie a,b,c. So, eg if we take a we get pa^2 + qa + r = a , ie pa^2 +a(q-1) + r = 0 . In all a,b,c we get same expression. So I think that means a , b ,c will be roots of px^2 + x(q-1) + r.
     
  5. Apr 9, 2013 #4

    Mark44

    Staff: Mentor

    The remainder is what?
    This is NOT given. It does NOT say that p(x) is divisible by x - a, or x - b, or x - c.
     
  6. Apr 9, 2013 #5

    ack

    User Avatar

    Id like you to consider this eg: let p(x)=x and when you divide this by x-a,x-b,x-c you get a,b,c as remainders and when you divide this by their combined product you get x ie p(x) itself as the remainder.So I think your remainder will be p(x) in your question ie the degree of p(x) is < 3.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted