中四數學問題,請教

請問:

let f(x) be a polynomial.when x^3+x^2-3x-4 is divided by f(x),the remainder is -x+4,find f(x)

a. x-2.

b. x+1

c.x^2-3x+5

d.x^2+3x+4

2 個解答

評分
  • 5 年 前
    最佳解答

    Let x³ + x² - 3x - 4 = p(x) f(x) - x + 4 , where f(x) is a polynomial of degree > 1 and not more than 3 , then x³ + x² - 2x - 8 = p(x) f(x).Note that p(2) f(2) = (2)³ + (2)² - 2(2) - 8 = 0 ,

    by remainder theorem, x - 2 is a factor of x³ + x² - 2x - 8.

    x³ + x² - 2x - 8

    = x³ - 8 + x(x - 2)

    = (x - 2)(x² + 2x + 4) + x(x - 2)

    = (x - 2)(x² + 3x + 4)∴ p(x) = x - 2 , f(x) = x² + 3x + 4 , the answer is D.

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  • 5 年 前

    由於 remainder 是 polynomial of degree 1

    你可以考慮 divisor 是 polynomial of degree larger than 1

    a. 和 b. 可以被否決。

    再考慮 (dividend - remainder) 要可以被 divisor f(x) 整除,你就可以知道是 c. 和 d. 之中的哪一個。

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