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Thread: finding P(x) and the remainder

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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; Jan 14th 2009 at 02:00 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 :
    $\displaystyle P(x)=(x-3)Q(x)-5$
    Now let $\displaystyle x=3$
    This gives $\displaystyle P(3)=0-5=-5$

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


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

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


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

    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 :
    $\displaystyle S(x)=ax+b$


    Now the previous working gives you the following system :
    $\displaystyle \left\{\begin{array}{ll} 3a+b=-5 \\ -2a+b=3 \end{array} \right.$
    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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    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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