F.4數學題...

f(x) = 2x^3 + x^2 + px + q
f(x) is divisible by 2x - 1

when f(x) is divided by 2x+3, remainder = 21
when f(x) is divided by (x-a), the remainder is f(a)
f(x) is divisible by 2x - 1
this means
f(x) is divisible by (x - 1/2)
f(1/2)=0
2(1/2)^3 + (1/2)^2 + p(1/2) + q=0
1/4+1/4+p/2+q=0
1+p+2q=0
p=(-1-2q)...(1)
when f(x) is divided by 2x+3, remainder = 21
this means
when f(x) is divided by (x+3/2), remainder = 21
f(3/2)=21
2(-3/2)^3 + (-3/2)^2 + p(-3/2) + q=21
2(-27/8)+(9/4)-3p/2+q=21
-9/2-3p/2+q=21
-9-3p+2q=42
-3p+2q=51...(2)
sub (1) into (2)
-3(-1-2q)+2q=51
8q=48
q=6
p=-1-2q=-1-2(6)=-13

2007-02-03 18:43:00 補充：
應是f(-3/2)=21

since f(x) is divisible by 2x-1,the remainder is 0.
so,f(0.5)=0
2(0.5)^3+(0.5)^2+p(0.5)+q=0
0.25+0.25+0.5p+q=0
0.5+0.5p+q=0
0.5p+q=-0.5...................(1)

when f(x) is divided by 2x+3,the remainder is 21.
so,f(-1.5)=21
2(-1.5)^3+(-1.5)^2+p(-1.5)+q=21
-6.75+2.25+(-1.5p)+q=21
-4.5-1.5p+q=21
-1.5p+q=25.5..................(2)

(1)+(2), (0.5p+q)+(-1.5p+q)=-0.5+25.5
0.5p+q-1.5p+q=25
-p+2q=25
p=-(25/2q)............................(3)

sub p=-(25/2q) into (2), -1.5(-25/2q)+q=25.5
18.75q+q=25.5
19.75q=25.5
q=1.36
sub q=1.36 into (3)

p=-[(25/2(1.36)]
p=17

so . p=17,q=1.36


i am not sure that the answers are right absolutely
but the method is totally right