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Math Help - Remainder Theorem

  1. #1
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    Exclamation Remainder Theorem

    Given that x^5 + ax^3 + bx^2 - 3 = (x^2 - 1)Q(x) - x - 2, where Q(x) is a polynomial.
    State the degree of Q(x) and find the value of a and of b.

    (I am able to deduce the degree i.e. 3
    I am also able to find the value of a ,i.e -2 and the value of b, i.e. 1.)


    However, the following question linked to the above is what I am not able to solve:

    Find also the remainder when Q(x) is divided by x + 2.

    (The answer given is -5.)
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Ilsa View Post
    Given that x^5 + ax^3 + bx^2 - 3 = (x^2 - 1)Q(x) - x - 2, where Q(x) is a polynomial.
    State the degree of Q(x) and find the value of a and of b.

    (I am able to deduce the degree i.e. 3
    I am also able to find the value of a ,i.e -2 and the value of b, i.e. 1.)


    However, the following question linked to the above is what I am not able to solve:

    Find also the remainder when Q(x) is divided by x + 2.

    (The answer given is -5.)
    Hint: Given a polynomial P(x), when you divide it by x - r, the remainder is equal to P(r).

    -Dan
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  3. #3
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    How do we get the answer -5 then?
    If the remainder is Q(-2), how do I further calculate the remainder?
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  4. #4
    Senior Member Sambit's Avatar
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    Let's proceed step by step.

    (i) Since Q(x) is of degree 3, assume Q(x)=Ax^3+Bx^2+Cx+D.

    (ii) Put this in RHS. Then by comparing the coefficients of RHS to the coefficients of LHS find the values of A,B,C and D.

    (iii) After getting the values of A,B,C,D you know the exact expression of Q(x). So you can easily put -2 in place of x and calculate Q(-2).
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  5. #5
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    That, actually finding Q, seems to me a very difficult way to do it. Topsquark's suggestion is best. You are told that
    x^5 + ax^3 + bx^2 - 3 = (x^2 - 1)Q(x) - x - 2 and you say you have determined that a= -2 and b= 1 so you have x^5 - 2x^3 + x^2 - 3 = (x^2 - 1)Q(x) - x - 2.

    Setting x= -2, (-2)^5- 2(-2)^3+ (-2)^2- 3= ((-2)^2- 1)Q(-2)- (-2)- 2

    -32+ 16+ 4- 3= -15= 3Q(-2)
    so it is easy to find Q(-2) which is, as Topsquark said, the remainder when Q(x) is divided by x+ 2.
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