數學多項式(很多字母)(一題而已幫忙@@

2009-08-16 9:23 pm
◎若( x^4 + 2x^3 + 3x^2 + 4x + 5 )( 5x^3 + 4x^2 +3x + 2)= ax^7 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + gx + h,則:


(1) a + b + c + d + e + f + g + h =?

(2) a - b + c - d + e - f + g - h =?

(3) a + c + e + g =?

(4) b + d + f + h =?
更新1:

◎如圖,矩形ABCD中,長AB線=3x+2,寬AD線=2x-1,若將AD線摺疊到 AB線上,使得D與F重合,則矩形BCEF面積為何? 圖:【ADEF是一個正方形E點和A點連一條線,ECFB是長方形,像我畫的這 樣,很難畫= =】 D───E───C |   |   | |───────| A   F   B

回答 (5)

2009-08-16 10:19 pm
✔ 最佳答案
◎若 (x4 + 2x3 + 3x2 + 4x + 5)(5x3 + 4x2 +3x + 2) = ax7 + bx6 + cx5 + dx4 + ex3 + fx2 + gx + h,則:
(1) a + b + c + d + e + f + g + h =?

ax7 + bx6 + cx5 + dx4 + ex3 + fx2 + gx + h
= (x4 + 2x3 + 3x2 + 4x + 5)(5x3 + 4x2 +3x + 2)

令 x = 1:
a + b + c + d + e + f + g + h
= (1 + 2 + 3 + 4 + 5)(5 + 4 + 3 + 2)
= 15 x 14
= 210

(2) a - b + c - d + e - f + g - h =?

ax7 + bx6 + cx5 + dx4 + ex3 + fx2 + gx + h
= (x4 + 2x3 + 3x2 + 4x + 5)(5x3 + 4x3 +3x + 2)

令 x = -1:
-a + b - c + d - e + f - g + h
= (1 - 2 + 3 - 4 + 5)(-5 + 4 - 3 + 2)
= 3 x (-2)
= -6

所以,a - b + c - d + e - f + g - h = 6

(3) a + c + e + g =?

a + b + c + d + e + f + g + h = 210 ... (*)
a - b + c - d + e - f + g - h = 6 ... (#)

(*) + (#):
2a + 2c + 2e + 2g = 216
所以, a + c + e + g = 108

(4) b + d + f + h =?

a + b + c + d + e + f + g + h = 210 ... (*)
a - b + c - d + e - f + g - h = 6 ... (#)

(*) - (#):
2b + 2d + 2f + 2h = 204
所以, b + d + f + h = 102


◎如圖,矩形ABCD中,長AB線=3x+2,寬AD線=2x-1,若將AD線摺疊到
AB線上,使得D與F重合,則矩形BCEF面積為何?
D───E───C
|   |   |
|───────|
A   F   B

BCEF 面積
= BC x FB
= (2x - 1)[(3x + 2) - (2x - 1)]
= (2x - 1)(x + 3)
= 2x2 + 6x - x - 3)
= 2x2 + 5x - 3

2009-08-16 18:00:54 補充:
第一題:
若把乘式展開,可得:
a = 5, b = 14, c = 26, d = 40, e = 54, f = 38, g = 23, = 10
「小儀子」計算 g 時錯了,導致 (1), (2), (3) 錯了。

2009-08-16 18:04:03 補充:
To tarcor:

第一題:g = 23 (不是 14)

第二題:常數項是 -3 (不是 +3)
2009-08-16 10:46 pm
可是我算出來的答案跟用速解法算的不一樣耶
a=5,b=14,c=26,d=40,e=54,f=38,g=14,h=10
是哪裡出了問題阿?

第二題矩形ECBF面積
[(3X+2)-(2X-1)]*(2X-1)
=2X^2+5X+3

2009-08-27 15:34:49 補充:
了解!!謝謝。
2009-08-16 10:34 pm
<1>ax^7 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + gx + h
a=5, b=14,c=26,d=40,e=54,f=38,g=15,h=10
(1)202
(2)-2
(3)100
(4)102
<2>
AF=AD=2x-1所以BF=AB-AF=>(3x+2)-(2x-1)=3x+2-2x+1=x+3
又AD=BC=2x-1
所以矩形BCEF面積=(2x-1)(x+3)=2x^2+5x-3
2009-08-16 10:03 pm
1.
f(1)
2.
f(-1)
3.
[f(1)-f(-1)]/2
4.
[f(1)+f(-1)]/2


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