Results 1 to 5 of 5

Math Help - Remainder theorem problem

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    22

    Remainder theorem problem

    Im currently stuck on this question:

    When P(z) is divided by z+1 the remainder is -8 and when divided by z-3 the remainder is 4. Find the remainder when P(z) is divided by (z-3)(z+1)

    I know that P(-1) = -8 and P(3) = 4 but i have no clue what to do from here.

    Thanks for any help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2009
    Posts
    68
    We have P(z)=(z+1)Q(z)-8.

    Let Q(z)=(z-3)R(z)+k.

    Thus 4=P(3)=4Q(3)-8=4k-8 \implies k=3.

    Hence P(z)=(z+1)(z-3)R(z)+3(z+1)-8=(z+1)(z-3)R(z)+3z-5 so the remainder when P(z) is divided by (z+1)(z-3) is 3z-5.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    22
    Im not sure what youve done here

    Thus

    And whats R(z)?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2009
    Posts
    68
    I am basically using the result that given any polynomials f(x),g(x) with g(x)\not\equiv0, there exist polynomials q(x),r(x) with either r(x)\equiv0 or \deg r(x)<\deg g(x) such that f(x)=q(x)g(x)+r(x). (This applies to polynomials over a field such as the field of rationals, reals or complex numbers.) The polynomial r(x) is the remainder when f(x) is divided by g(x).

    Actually, I didn’t do a great job in my post above. Let me re-do the proof in a clearer way.

    Given polynomials P(z) and (z-1)(z+3) we have that there exist a polynomial R(z) and constants k,h such that

    P(z)\ =\ (z-1)(z+3)R(z)+kx+h

    As P(-1)=-8 and P(3)=4 we have -k+h=-8 and 3k+h=4. Solving the two simultaneous equations should yield k=3,\,h=-5. Hence the remainder is 3x-5.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Sep 2009
    Posts
    22
    Ahh i see :d

    Thanks very much for your help
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basic Chinese Remainder Theorem Problem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: May 22nd 2010, 09:27 AM
  2. Chinese Remainder Theorem Problem
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 21st 2009, 01:49 PM
  3. The Remainder Estimation Theorem Problem
    Posted in the Calculus Forum
    Replies: 0
    Last Post: April 28th 2008, 04:26 PM
  4. The Remainder Estimation Theorem Problem
    Posted in the Calculus Forum
    Replies: 14
    Last Post: April 27th 2008, 02:07 PM
  5. Factor Theorem and Remainder Theorem
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 8th 2007, 10:50 AM

Search Tags


/mathhelpforum @mathhelpforum