中四數學~

1.Let f(x)=x^3+6x^2+px+6.When f(x) is divided by x+2andx+3 respectiely,the remainder odtained in th second one is larger than that obtained in the first one by 7.Find p.

the remainder when f(x) is divided by x+2
= f(-2) = (-2)^3 + 6 (-2)^2 + p(-2) + 6
= -8 + 24 - 2p + 6
= -2p + 22
the remainder when f(x) is divided by x+3
= f(-3) = (-3)^3 + 6 (-3)^2 + p(-3) + 6
= -27 + 54 - 3p + 6
= -3p + 33

as yhe remainder odtained in th second one is larger than that obtained in the first one by 7
so -3p + 33 = -2p + 22 + 7
33 - 22 - 7 = p
hence p = 4

2.When a polynomial P(x) is divided by x+1 and x-3,the remainders are 1 and -3 respectively.Find the remainder when P(x) is divided by (x+1)(x-3).

let the remainder when P(x) is divided by (x+1)(x-3) be ax + b, and Q(x) be the quotient
So P(x) = Q(x)(x+1)(x-3) + ax + b
when P(x) is divided by x+1 the remainder = 1
so P(-1) = Q(-1)(-1+1)(-1-3) + a(-1) + b = -a + b = 1 -- (1)
when P(x) is divided by x-3 the remainder = -3
so P(3) = Q(3)(3+1)(3-3) + a(3) + b = 3a + b = -3 -- (2)

(2) - (1):
3a + b - (-a) - b = -3 - 1
4a = -4
a = -1 -- (3)
Put (3) into 1
-(-1) + b = 1
b = 0

so the remainder when P(x) is divided by (x+1)(x-3) is -x

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1. let the remainder odtained in the first one be a
so the remainder odtained in the second one is a+7
f(-2)=(-2)³+6(-2)²-2p+6
a=-2+24-2p+6
a=28-2p------------(1)
f(-3)=(-3)³+6(-3)²-3p+6
a+7=-27+54-3p+6
a=26-3p------------(2)
solving (1) and (2)
we get p=-2 a=32

2. let the remainder of P(x) divided by (x+1)(x-3) be mx+n
P(x)=(x+1)(x-3)R(x)+mx+n , where R(x) is some polynomial
P(-1)=-m+n
1=-m+n---------(3)
P(3)=3m+n
-3=3m+n--------(4)
solving (3)and(4)
we get m=-1 , n=0
so the remainder when P(x) is divided by (x+1)(x-3) is -x

2007-12-30 19:20:16 補充：
sorry for some mistakes1. f(-2)=(-2)³+6(-2)²-2p+6a=-8+24-2p+6a=22-2p-----(1)and a=26-3p------------(2)solving , we get a=14,p=4so p=4