Given that the expression 2x^3 + mx^2 + nx + c leaves the same remainder when divided by x -2 or by x+1 I prove that m+n =-6

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Given that the expression 2x^3 + mx^2 + nx + c leaves the same remainder when divided by x -2 or by x+1 I prove that m+n =-6

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Hải Đăng 3 years 2021-08-07T07:44:45+00:00 1 Answers 178 views 0

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  1. Sigrid
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    2021-08-07T07:46:09+00:00
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    Given:

    The expression is:

    2x^3+mx^2+nx+c

    It leaves the same remainder when divided by x -2 or by x+1.

    To prove:

    m+n=-6

    Solution:

    Remainder theorem: If a polynomial P(x) is divided by (x-c), thent he remainder is P(c).

    Let the given polynomial is:

    P(x)=2x^3+mx^2+nx+c

    It leaves the same remainder when divided by x -2 or by x+1. By using remainder theorem, we can say that

    P(2)=P(-1)              …(i)

    Substituting x=-1 in the given polynomial.

    P(-1)=2(-1)^3+m(-1)^2+n(-1)+c

    P(-1)=-2+m-n+c

    Substituting x=2 in the given polynomial.

    P(2)=2(2)^3+m(2)^2+n(2)+c

    P(2)=2(8)+m(4)+2n+c

    P(2)=16+4m+2n+c

    Now, substitute the values of P(2) and P(-1) in (i), we get

    16+4m+2n+c=-2+m-n+c

    16+4m+2n+c+2-m+n-c=0

    18+3m+3n=0

    3m+3n=-18

    Divide both sides by 3.

    \dfrac{3m+3n}{3}=\dfrac{-18}{3}

    m+n=-6

    Hence proved.

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Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )