polynomials question

👨🏻‍🏫 學術台數學

恩樂

2008-02-02 11:59 pm

f(x) = ax^3 - 13x^2 + bx - 60 除以 3x^2 - 14x - 5 時 , 餘式是 7x - 35

求 a 及 b

更新1:

(i) 求 a 及 b (ii) 因式分解 f(x) (b) 已知 g(x) = 9x^3 + 60x^2 + 28x - 48 與 f(x) 有公因式 (i) 因式分解 g(x) (ii) 解 3f(x) = 2g(x) (iii) 解 2f(x) = 3g(x) 還想請教大家 , 有冇D技巧可以較輕鬆處理運算量大既題目 ?

回答 (1)

[匿名]

2008-02-03 12:23 am

✔ 最佳答案

ax^3 - 13x^2 + bx - 60 -(7x - 35)
= ax^3 - 13x^2 + (b-7)x - 25
Assume (3x^2 - 14x - 5) X (cx + d) = ax^3 - 13x^2 + (b-7)x - 25
So, d = 5 for -5 X 5 = -25
(3x^2 - 14x - 5) X (cx + 5)
= (3cx^3 - 14cx^2 - 5cx) + (15x^2 - 70x - 25)
= 3cx^3 + 1cx^2 - (5c + 70)x - 25

We then get the following:
3c = a
c = -13
- (5c + 70) = b-7 or b-7 = -15c -70
i.e. b = 7 -15c -70
put c=-13
b = 7 - 15X-13 -70
= 132

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