求各位數學高手指導

(a)
The highest possible degree of R(x) = 1

(The degree of (x - 1)(x + 1) is 2. If the degree of R(x) is 2 or higher, R(x)can be further divided by (x - 1)(x + 1).)


(b)
Let Q(x) and (ax + b) be the quotient and R(x) when P(x) is divided by (x - 1)(x+ 1).
Then, P(x) = (x - 1)(x + 1)•Q(x) + (ax + b)

When P(x) is divided by (x - 1), the remainder = -2.
P(1) = -2
a(1) + b = -2
a + b = -2 ...... [1]

When P(x) is divided by (x + 1), the remainder = 6.
P(-1) = 6
a(-1) + b = -2
-a + b = 6 ...... [2]

[1] - [2] :
2a = -8
a = -4

[1] + [2] :
2b = 4
b = 2

R(x) = -4x + 2

柏楷 ( ) 並沒回答問題，只示範何謂 copy and paste。

(a)
The highest possible degree of R(x) = 1

(The degree of (x - 1)(x + 1) is 2. If the degree of R(x) is 2 or higher, R(x)can be further divided by (x - 1)(x + 1).)


(b)
Let Q(x) and (ax + b) be the quotient and R(x) when P(x) is divided by (x - 1)(x+ 1).
Then, P(x) = (x - 1)(x + 1)•Q(x) + (ax + b)

When P(x) is divided by (x - 1), the remainder = -2.
P(1) = -2
a(1) + b = -2
a + b = -2 ...... [1]

When P(x) is divided by (x + 1), the remainder = 6.
P(-1) = 6
a(-1) + b = -2
-a + b = 6 ...... [2]

[1] - [2] :
2a = -8
a = -4

[1] + [2] :
2b = 4
b = 2

R(x) = -4x + 2

(b)
P(x)=Q(x)(x－1)(x+1)+a(x－1)－2
P(－1)=a*(－2)－2=6
a=－4
R(x)=－4(x－1)－2=－4x+2