Ray
Lv 6
Ray asked in 科學及數學數學 · 1 decade ago

F4 Maths Dividing Polynomials2

In each question, divide A by B, and write the relationship between the two.

1. A=x^4+6x^3-7x^2+3x+9 B=x^2+3x-2

2. A=x^3+6x^2+6x-6 B=x^2+3x-3

2 Answers

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  • wkho28
    Lv 7
    1 decade ago
    Favorite Answer

    1. A=x^4+6x^3-7x^2+3x+9 B=x^2+3x-2

        1+3-14

       ----------------------------

    1+3-2 ) 1+6-7+3+9

        1+3-2

        ----------------

         3-5+3

         3+9-6

        --------------------

          -14 +9 + 9

          -14-42+28

         --------------------

            51-19

    Quotient = x^2+3x-14

    Remainder = 51x-19

    (x^4+6x^3-7x^2+3x+9) = (x^2+3x-14) (x^2+3x-2)+51x-19

    2. A=x^3+6x^2+6x-6 B=x^2+3x-3

        1+3

       ---------------------

    1+3-3 ) 1+6+6-6

        1+3-3

        --------------

         3+9-6

         3+9-9

        -------------

           3

    Quotient = x+3

    Remainder = 3

    x^3+6x^2+6x-6 = (x+3) (x^2+3x-3)+3

    Alternative method :

    Let x^3+6x^2+6x-6≡(Px+Q) (x^2+3x-3)+Rx+S

    x^3+6x^2+6x-6≡Px^3+3Px^2-3Px+Qx^2+3Qx-3Q+Rx+S

    x^3+6x^2+6x-6≡Px^3+(3P+Q)x^2-(3P-3Q-R)x-3Q+S

    Compare the coefficients :

    P=1 . . . . . . . . . . . (1)

    3P+Q=6 . . . . . . . . (2)

    -(3P-3Q-R)=6

    -3P+3Q+R=6 . . . . (3)

    -3Q+S=-6 . . . . . . . (4)

    Put (1) into (2) : 3+Q=6

    Q=3

    Put P=1 and Q=3 in to (3) : -3+9+R=6

    R=0

    Put Q=3 into (4) : -9+S=-6

    S=3

    Therefore x^3+6x^2+6x-6≡(x+3) (x^2+3x-3)+3

    2009-09-28 09:02:12 補充:

    It is quite obvious that :

    In Q1,

    x^4+6x^3-7x^2+3x+9 = (x^2+3x-14) (x^2+3x-2)+51x-19

    A = (x^2+3x-14) B + 51x - 19

    In Q2,

    x^3+6x^2+6x-6 = (x+3) (x^2+3x-3)+3

    A = (x+3) B + 3

    2009-09-28 09:27:42 補充:

    In Q1,

    x^4+6x^3-7x^2+3x+9 = (x^2+3x-14) (x^2+3x-2)+51x-19

    A = (x^2+3x-14) B + 51x - 19

    In Q2,

    x^3+6x^2+6x-6 = (x+3) (x^2+3x-3)+3

    A = (x+3) B + 3

  • 1 decade ago

    Is it no problem for Q1 ?

    2009-09-27 23:20:52 補充:

    wkho28 :

    You haven't write the relationship between A and B.

    I think Q1 have some word mistake.

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