? 發問於 科學及數學數學 · 1 十年 前

Two questions on polynomial (F.4)

1.When a polynomial f(x) is divided by x-5,the remainder is 9.When it is divided by x+2,the remainder is -5.Find the remainder when f(x) is divided by (X-5)(x+2).

2.Let f(x) =3x^3 +mx^2 -nx -7 .When f(x) is divided by (x+1)(x-3),the remainder is 2x-4.

(a)Find the values of f(-1) and f(3).

(b)Set up two equations connecting m and n.

(c)Find the values of m and n.

2 個解答

評分
  • 1 十年 前
    最佳解答

    1.When a polynomial f(x) is divided by x-5,the remainder is 9.When it is divided by x+2,the remainder is -5.Find the remainder when f(x) is divided by (x-5)(x+2).

    SOLUTION

    let the remainder is ax+b

    then

    f(x)=g(x)(x-5)(x+2)+(ax+b)

    Then by remainder theorem

    f(5)=5a+b=9...(1)

    f(-2)=-2a+b=-5...(2)

    (1)-(2):

    7a=14

    a=2

    sub into (2) b=2a-5=-1

    So the remainder when f(x) is divided by (x-5)(x+2) is

    2x-1

    2.Let f(x) =3x^3 +mx^2 -nx -7 .When f(x) is divided by (x+1)(x-3),the remainder is 2x-4.

    (a)Find the values of f(-1) and f(3).

    (b)Set up two equations connecting m and n.

    (c)Find the values of m and n.

    SOLUTION

    (a)

    We have

    f(x)=g(x)(x+1)(x-3)+2x-4

    f(-1)=2(-1)-4=-6

    f(3)=2(3)-4=2

    (b)

    Since f(-1)=-6

    f(-1)=3(-1)^3 +m(-1)^2 -n(-1) -7=-6

    -3+m+n-7=-6

    m+n=4...(1)

    This is the first equation

    Since f(3)=2

    f(3)=3(3)^3 +m(3)^2 -n(3) -7=2

    81+9m-3n-7=2

    9m-3n=-72

    3m-n=-24

    n-3m=24...(2)

    This is the second equation

    (c)

    from (1)

    m=4-n

    sub into (2)

    n-3(4-n)=24

    -12+4n=24

    n=9

    m=4-9=-5

    2007-04-18 00:33:48 補充:

    答題中的 g(x) 就是 quotinet.應該用Q(x) 會好看些﹐不過其實都是同一樣東西。

  • 1 十年 前

    1)

    Assume the remainder be Ax + B where A and B are constants.

    i.e. f(x) = Q (x-5)(x+2) + (Ax + B)

    As given, f(5) = 9 and f(-2) = -5

    f(5) = 5A + B = 9 --- (1)

    f(-2) = -2A + B = -5 --- (2)

    (1) - (2)

    7A = 14

    A = 2

    Put it back into (1)

    5(2) + B = 9

    B = -1

    So, remainder is 2x - 1

    **********************

    2)

    a) Assume f(x) = Q(x+1)(x-3) + (2x-4) where Q is the quotient.

    f(-1) = 2(-1) - 4 = -6

    f(3) = 2(3) - 4 = 2

    b) f(x) = 3x^3 + mx^2 - nx - 7

    f(-1) = 3(-1)^3 + m(-1)^2 - n(-1) - 7 = -6

    -3 + m + n - 7 = -6

    m + n = 4 --- (1)

    f(3) = 3(3)^3 + m(3)^2 - n(3) - 7 = 2

    81 + 9m - 3n - 7 = 2

    3m - n = -24 --- (2)

    c)

    (1) + (2)

    4m = -20

    m = -5

    n = 9

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