Polynominal problem

[匿名]

2015-08-02 2:49 pm

Please help me to solve this problem.

f(x) is a polynomial of degree 3 and f(1) = 2, f(-2) = -7.
i) Find the remainder if f(x) is divided by x^2 + x - 2.
ii) Find f(x) if f(x) is divisible by 1-3x and f(0) = 5.

回答 (1)

邊位都好

2015-08-02 5:16 pm

✔ 最佳答案

(i) Let Ax＋B and Q(x) be the remainder and the quotientwhen f(x) is divided by x²＋x－2, so,f(x)＝(x²＋x－2)Q(x)＋Ax＋Bf(1)＝A＋B＝2⋯⋯⋯⋯⋯⋯(1)f(-2)＝-2A＋B＝-7⋯⋯⋯⋯(2)(1)－(2) get A＝3Sub. into (1) get B＝-1∴ the remainder is 3x－1
(ii) f(x)＝(x²＋x－2)Q(x)＋3x－1 ＝(x²＋x－2)Q(x)－(1－3x)As f(x) is divisible by 1－3x, so, Q(x) is 1－3x∴ f(x)＝k(x²＋x－2)(1－3x)－(1－3x) ⋯⋯ where k is a constantf(0)＝5==> k(-2)(1)－1＝5==> k＝-3∴ f(x)＝-3(x²＋x－2)(1－3x)－(1－3x) ＝(3x－1)(3x²＋3x－5)

2015-08-02 09:19:59 補充：
so, Q(x) is k(1－3x) ⋯⋯ (missing the letter "k", sorry)

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