Form4 math

1.Let f(x) = px3﹣13x2﹣14x+21
f(-3/2) = 6
p(-3/2)3﹣13(-3/2)2﹣14(-3/2)+21 = 6
-27p/8﹣117/4+21+21 = 6
-27p/8 = -27/4
-27p = -54
p = 2

2.Let f(x) = 3x3+px2+86x﹣191
f(7) = p
3(7)3+p(7)2+86(7)﹣191 = p
1029+49p+602﹣191 = p
1440+49p = p
48p = -1440
p = -30

3.Let f(x) = x5000﹣x1001+1
the remainder
= f(-1)
= (-1)5000﹣(-1)1001+1
= 1﹣(-1)+1
= 1+1+1
= 3

4.when 2x2+25x+80 is divided by x+q??
Actually I've answered the same question. Please refer to:
http://hk.knowledge.yahoo.com/question/?qid=7007123004556
Let f(x) = 2x2+25x+80
f(-q) = q
2(-q)2+25(-q)+80 = q
2q2﹣25q+80 = q
2q2﹣26q+80 = 0
q2﹣13q+40 = 0
(q﹣5)(q﹣8) = 0
q = 5 or q = 8

5.Let f(x) = 2qx2+64x﹣14
f(1/q) = -q/2
2q(1/q)2+64(1/q)﹣14 = -q/2
2q/q2+64/q﹣14 = -q/2
(2q+64q﹣14q2)/q2 = -q/2
2(66q﹣14q2) = -q3
132q﹣28q2 = -q3
q3﹣28q2+132q = 0
q(q2﹣28q+132) = 0
q(q﹣22)(q﹣6) = 0
q = 0 or q = 22 or q = 6

6.f(x) = px3+5x2﹣x﹣q
f(-1) = 2
p(-1)3+5(-1)2﹣(-1)﹣q = 2
-p+5+1﹣q = 2
-p﹣q = -4
p+q = 4 ---(1)
f(1) = 6
p(1)3+5(1)2﹣1﹣q = 6
p+5﹣1﹣q = 6
p﹣q = 2 ---(2)
(1)+(2):
2p = 6
p = 3 ---(3)
Put (3) into (2),
3﹣q = 2
q = 1
∴p = 3,q = 1

1.Let f(x) = px3﹣13x2﹣14x+21
f(-3/2) = 6
p(-3/2)3﹣13(-3/2)2﹣14(-3/2)+21 = 6
-27p/8﹣117/4+21+21 = 6
-27p/8 = -27/4
-27p = -54
p = 2

2.Let f(x) = 3x3+px2+86x﹣191
f(7) = p
3(7)3+p(7)2+86(7)﹣191 = p
1029+49p+602﹣191 = p
1440+49p = p
48p = -1440
p = -30

3.Let f(x) = 2x2+25x+80
f(-q) = q
2(-q)2+25(-q)+80 = q
2q2﹣25q+80 = q
2q2﹣26q+80 = 0
q2﹣13q+40 = 0
(q﹣5)(q﹣8) = 0
q = 5 or q = 8

4.Let f(x) = 2qx2+64x﹣14
f(1/q) = -q/2
2q(1/q)2+64(1/q)﹣14 = -q/2
2q/q2+64/q﹣14 = -q/2
(2q+64q﹣14q2)/q2 = -q/2
2(66q﹣14q2) = -q3
132q﹣28q2 = -q3
q3﹣28q2+132q = 0
q(q2﹣28q+132) = 0
q(q﹣22)(q﹣6) = 0
q = 0 or q = 22 or q = 6

5.f(x) = px3+5x2﹣x﹣q
f(-1) = 2
p(-1)3+5(-1)2﹣(-1)﹣q = 2
-p+5+1﹣q = 2
-p﹣q = -4
p+q = 4 ---(1)
f(1) = 6
p(1)3+5(1)2﹣1﹣q = 6
p+5﹣1﹣q = 6
p﹣q = 2 ---(2)
(1)+(2):
2p = 6
p = 3 ---(3)
Put (3) into (2),
3﹣q = 2
q = 1
∴p = 3,q = 1
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1.when px^3-13x^2-14x+21 is divided by 2x+3,the remainder is 6.
let f(x)=px^3-13x^2-14x+21
f(-3/2)=p(-3/2)^3-13(-3/2)^2-14(-3/2)+21=6
-(27/8)p-13(9/4)+14(3/2)+21=6
-(27/8)p-117/4+21+21=6
-(27/8)p=-27/4
p=2

2.when 3x^3+px^2+86x-191 is divided by x-7,the remainder is p

let f(x)=3x^3+px^2+86x-191
f(7)= 3(7)^3+p(7)^2+86(7)-191=p
1029+49p+602-191=p
48p=-1440
p=-30

3.when 2x^2+25x+80 is divided by x+q,the remainder is q

let f(x)=2x^2+25x+80
f(-q)=2(-q)^2+25(-q)+80=q
2q^2-25q+80=q
2q^2-26q+80=0
q^2-13q+40=0
(q-8)(q-5)=0
q=8 or q=5

4.when 2qx^2+64x-14 is divided by qx-1,the remainder is -q/2

let f(x)=2qx^2+64x-14
f(1/q)=2q(1/q)^2+64(1/q)-14=-q/2
2/q+64/q-14=-q/2
whole equaton x 2q
4+128-28q=-q^2
q^2-28q+132=0
(q-22)(q-6)=0
q=22 or q=6

5.let f(x)=px^3+5x^2-x-q.when f(x) is divided by x+1 and x-1,the remainders are 2 and 6 respectively.

f(x)=px^3+5x^2-x-q
f(-1)=p(-1)^3+5(-1)^2-(-1)-q=2
f(1)=p(1)^3+5(1)^2-(1)-q=6

-p+5+1-q=2
p+5-1-q=6

add two equations
10-2q=8
q=1
p=3

1.
p(-(3/2))³-13(-(3/2))²-14(-(3/2))+21=6, Solution is: 2

2.
3(7)³+p(7)²+86(7)-191=p, Solution is: -30

3.
(-1)⁵⁰⁰⁰-(-1)¹⁰⁰¹+1= 3

4.
syntax error

5.
2q((1/q))²+64((1/q))-14=-(q/2), Solution is: 6,22

6.
p(-1)+5+1-q=2
p+5-1-q=6

p+q=4
p-q=2

p=3
q=1