中四數學more about polynomials

[匿名]

2011-03-20 8:18 pm

when a polynomial p(x) is divided by x+1 and x+2,the remainders are -7 and -25 respectively.Find the remainder when p(x) is divided by (x+1)(x+2).please show full steps.thank you

回答 (1)

用戶2053

2011-03-20 8:26 pm

✔ 最佳答案

Let the remainder when p(x) is divided by (x+1)(x+2) be ax + b :thenP(x) = Q(x) * (x+1) (x+2) + ax + bBy remainder theoream :P(- 1) = - 7
0 + a(-1) + b = - 7
a - b - 7 = 0 .....(1)P(- 2) = - 25
0 + a(-2) + b = - 25
2a - b - 25 = 0 .....(2)(2) - (1) :a - 25 + 7 = 0
a = 18
Sub into (1) :
18 - b - 7 = 0
b = 11Answer : 18x + 11

2011-03-20 12:31:30 補充：
Alternative :

P(x) = Q(x) * (x+1)(x+2) + k(x+1) - 7

P(-2) = 0 + k(-2+1) - 7

- 25 = - k - 7

k = 18

The remainder is k(x+1) - 7 = 18(x+1) - 7 = 18x + 11

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