F.4數學多項式

1)
4x^2-5x+11
/------------------------------------
x^2-3 / 4x^4 - 5x^3 -x^2 +0x +6
4x^4 -12x^2
-----------------------------------
-5x^3 +11x^2 +0x +6
-5x^3 +15x
-------------------------------
11x^2 -15x +6
11x^2 -33
------------------------
-15x+39
商 : 4x^2-5x+11
餘 : -15x-27
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2)至於短除法....我沒有聽過 不好意思唷
只能說答案是
商 : 4x^3-2x+9
餘 : -6
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3)
A(x-2)(x+3)+B(x-2)+C = 5x^2+2x-20
For x=2,
C=5(4)+2(2)-20=4
---------------------------
For x=-3,
B(-5)+C=5(9)+2(-3)-20
-5B+4=19
B=-3
--------------------------
For x=0,
A(-2)(3)+B(-2)+C=-20
-6A-2(-3)+4=-20
A=5
------------------------------
4)
y=ax^2+bx+c
When x=0, y=16
c=16 <------------------------------------
-------------------------
When x=9, y=7
7=81a+9b+16
81a+9b= -9
9a+b= -1 ------- (1)
When x=8, y=0
0=64a+8b+16
64a+8b= -16
8a+b= -2 ---------(2)

(1)-(2) : a=1 <-------------------------------
(1) : 9(1)+b=-1
b=-10 <----------------------------
--------------------------------------
5)
Let x^2+px+q=0 be the equation with roots √7+5 & √7-5
-p=(√7+5)+(√7-5)
p=-2√7
------------------
q=(√7+5)(√7-5)
=7-25=-18
x^2-2√7x-18=0 is the required equation.

2014-02-06 19:06:43 補充：
12x^4+4x^3-6x^2+25x+3 = (3x+1)Q(x)+R = 3[ (x-(-1/3))Q(x)+R/3 ]
4x^4+(4/3)x^3-2x^2+(25/3)x+1=(x-(-1/3))Q(x)+R/3
Synthetic division:
4 4/3 -2 25/3 1
-1/3 l -4/3 0 2/3 -3
-------------------------------
4 0 -2 9 -2
Q(x)=4x^3-2x+9
R/3=-2
R=-6

#5

根之和 = √7+5 + √7-5 = 2√7
根之積 = (√7+5)(√7-5) = 7-25 = -18

所求方程是 x² - 2√7 x - 18 = 0

2014-02-02 17:34:22 補充：
#4

y = ax² + bx + c 穿過 (8,0), (9,7), (0,16)

0 = 64a + 8b + c
7 = 81a + 9b + c
16 = 0 + 0 + c

c = 16

{64a + 8b = -16
{81a + 9b = -9

{8a + b = -2
{9a + b = -1

a = 1
b = -10

2014-02-02 17:36:23 補充：
#3

A(x - 2)(x + 3) + B(x - 2) + C ≡ 5x² + 2x -20

代 x = 2，C = 20+4-20 = 4
代 x = -3，-5B + C = 45 - 6 -20

-5B + 4 = 19
-5B = 15
B = -3

比較 x² 係數： A = 5

2014-02-02 17:39:47 補充：
#2 和 #1

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2014-02-02 17:57:25 補充：
哈哈，客氣啦～

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