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Math Help - finding P(x) and the remainder

  1. #1
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    finding P(x) and the remainder

    If P(x) has the remainder -5 when divided by x-3 and the remainder 3 when divided by x+2, find the reminder when P(x) is divided by (x-3)(x+2)
    how do you find remainder from the details provided above?
    Last edited by mr fantastic; January 14th 2009 at 03:00 AM. Reason: Clarified post
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  2. #2
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    Imagine you write the euclidean division :

    There exists a polynomial Q such that :
    P(x)=(x-3)Q(x)-5
    Now let x=3
    This gives P(3)=0-5=-5

    Same thing for the division by (x+2) : we get P(-2)=3


    Now what if we consider the division by (x-3)(x+2) ?
    There exists polynomials R and S (S is the remainder) such that :
    P(X)=(x-3)(x+2)R(x)+S(x)
    Now once again, if you let x=3, you get :

    \underbrace{P(3)}_{-5}=0+S(3) so we have S(3)=-5
    Similarly, if you let x=-2, you get : S(-2)=3


    So the remainder of P when divided by (x-2)(x+3) will be a polynomial S such that S(3)=-5 and S(-2)=3

    But the degree of S cannot exceed 2, since (x-2)(x+3) has a degree 2. (if S has a degree equal or superior to 2, then it still can be divided by (x-2)(x+3))
    Hence we're looking for a and b in :
    S(x)=ax+b


    Now the previous working gives you the following system :
    \left\{\begin{array}{ll} 3a+b=-5 \\ -2a+b=3 \end{array} \right.
    Which is very basic algebra
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    how if P(x) was a cubic equation? is it still possible to find P(x)
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    Quote Originally Posted by hungrybarts View Post
    how if P(x) was a cubic equation? is it still possible to find P(x)
    You're too fast >< I've edited my post !
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  5. #5
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    Quote Originally Posted by hungrybarts View Post
    how do you find remainder from the details provided above?
    Thread of related interest: http://www.mathhelpforum.com/math-he...mial-help.html

    By the way, the question is asking for the remainder so I don't know why you keep talking about P(x). P(x) is not the remainder.
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