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Math Help - Dividing polynomials

  1. #1
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    Dividing polynomials

    If a polynomial is divided by (x+2), the remainder is -19. When the same polynomial is divided by (x-1), the remainder is 2. Determine the remainder when the polynomial is divided by (x-1)(x+2).
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  2. #2
    Junior Member
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    Gothenburg, Sweden
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    Hi!
    We know that p(x)=k_1 (x)(x+2)-19 and p(x)=k_2 (x)(x-1)+2 where k_1 (x) and k_2 (x) are two polynomials such that deg(k_1 (x))=deg(k_2(x)).
    Thus, p(1)=2 and p(-2)=-19.

    The remainer, when dividing p(x) by (x-1)(x+2) can be of degree 1 at most since (x-1)(x+2) is a polynomial of degree 2.
    This is, r(x)=ax+b and
    p(x)=k_3(x)(x-1)(x+2)+r(x)=k_3(x)(x-1)(x+2)+ax+b where k_3(x) is a polynomial of degree deg(k_1(x))-1

    We see that p(1)=a+b and p(-2)=-2a+b but we know that p(1)=2 and p(-2)=-19 so we've to solve a linear system of equations in the unknowns a and b. It's easy to solve this system and the solution is \begin{cases} -2a+b=-19 \\ a+b=2 \end{cases} \, \, \, \, \, \, \, \, \, \Longrightarrow \, \, \, \, \, \, \, \, \, \begin{cases} a=7 \\ b=-5 \end{cases}

    Therefore, the reaminder when p(x) is divided by (x-1)(x+2) is r(x)=7x-5
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