Answer :
To find the remainder when the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( x(x^2 - 1) \)[/tex], follow these detailed steps.
We know that:
1. When [tex]\( P(x) \)[/tex] is divided by [tex]\( x-1 \)[/tex], the remainder is 1.
2. When [tex]\( P(x) \)[/tex] is divided by [tex]\( x \)[/tex], the remainder is 2.
3. When [tex]\( P(x) \)[/tex] is divided by [tex]\( x+1 \)[/tex], the remainder is 3.
We need to express [tex]\( P(x) \)[/tex] in a form that helps us find the remainder when divided by [tex]\( x(x^2 - 1) \)[/tex].
A helpful approach is to write the polynomial [tex]\( P(x) \)[/tex] as:
[tex]\[ P(x) = (x-1)(x)(x+1)Q(x) + ax^2 + bx + c \][/tex]
where [tex]\( Q(x) \)[/tex] is some polynomial and [tex]\( ax^2 + bx + c \)[/tex] is the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( x(x^2 - 1) \)[/tex].
Given the remainders, we have three conditions from insertions into [tex]\( P(x) \)[/tex]:
1. [tex]\( P(1) = a(1)^2 + b(1) + c = 1 \)[/tex]
2. [tex]\( P(0) = c = 2 \)[/tex]
3. [tex]\( P(-1) = a(-1)^2 - b(1) + c = 3 \)[/tex]
Using these conditions, we can set up a system of equations:
1. [tex]\( a + b + c = 1 \)[/tex]
2. [tex]\( c = 2 \)[/tex]
3. [tex]\( a - b + c = 3 \)[/tex]
Starting with the second equation directly gives us:
[tex]\[ c = 2 \][/tex]
Substituting [tex]\( c = 2 \)[/tex] into the first and third equations, we get:
1. [tex]\( a + b + 2 = 1 \)[/tex]
2. [tex]\( a - b + 2 = 3 \)[/tex]
Simplifying these equations:
1. [tex]\( a + b + 2 = 1 \)[/tex] simplifies to [tex]\( a + b = -1 \)[/tex]
2. [tex]\( a - b + 2 = 3 \)[/tex] simplifies to [tex]\( a - b = 1 \)[/tex]
Now we have a simpler system of linear equations:
1. [tex]\( a + b = -1 \)[/tex]
2. [tex]\( a - b = 1 \)[/tex]
To solve this system, we can add the two equations:
[tex]\[ (a + b) + (a - b) = -1 + 1 \][/tex]
[tex]\[ 2a = 0 \][/tex]
[tex]\[ a = 0 \][/tex]
Next, substituting [tex]\( a = 0 \)[/tex] into the equation [tex]\( a + b = -1 \)[/tex]:
[tex]\[ 0 + b = -1 \][/tex]
[tex]\[ b = -1 \][/tex]
We have found that:
[tex]\[ a = 0 \][/tex]
[tex]\[ b = -1 \][/tex]
[tex]\[ c = 2 \][/tex]
Thus, the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( x(x^2 - 1) \)[/tex] is:
[tex]\[ 0x^2 - x + 2 = -x + 2 \][/tex]
Therefore, the remainder is:
[tex]\[ -x + 2 \][/tex]
We know that:
1. When [tex]\( P(x) \)[/tex] is divided by [tex]\( x-1 \)[/tex], the remainder is 1.
2. When [tex]\( P(x) \)[/tex] is divided by [tex]\( x \)[/tex], the remainder is 2.
3. When [tex]\( P(x) \)[/tex] is divided by [tex]\( x+1 \)[/tex], the remainder is 3.
We need to express [tex]\( P(x) \)[/tex] in a form that helps us find the remainder when divided by [tex]\( x(x^2 - 1) \)[/tex].
A helpful approach is to write the polynomial [tex]\( P(x) \)[/tex] as:
[tex]\[ P(x) = (x-1)(x)(x+1)Q(x) + ax^2 + bx + c \][/tex]
where [tex]\( Q(x) \)[/tex] is some polynomial and [tex]\( ax^2 + bx + c \)[/tex] is the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( x(x^2 - 1) \)[/tex].
Given the remainders, we have three conditions from insertions into [tex]\( P(x) \)[/tex]:
1. [tex]\( P(1) = a(1)^2 + b(1) + c = 1 \)[/tex]
2. [tex]\( P(0) = c = 2 \)[/tex]
3. [tex]\( P(-1) = a(-1)^2 - b(1) + c = 3 \)[/tex]
Using these conditions, we can set up a system of equations:
1. [tex]\( a + b + c = 1 \)[/tex]
2. [tex]\( c = 2 \)[/tex]
3. [tex]\( a - b + c = 3 \)[/tex]
Starting with the second equation directly gives us:
[tex]\[ c = 2 \][/tex]
Substituting [tex]\( c = 2 \)[/tex] into the first and third equations, we get:
1. [tex]\( a + b + 2 = 1 \)[/tex]
2. [tex]\( a - b + 2 = 3 \)[/tex]
Simplifying these equations:
1. [tex]\( a + b + 2 = 1 \)[/tex] simplifies to [tex]\( a + b = -1 \)[/tex]
2. [tex]\( a - b + 2 = 3 \)[/tex] simplifies to [tex]\( a - b = 1 \)[/tex]
Now we have a simpler system of linear equations:
1. [tex]\( a + b = -1 \)[/tex]
2. [tex]\( a - b = 1 \)[/tex]
To solve this system, we can add the two equations:
[tex]\[ (a + b) + (a - b) = -1 + 1 \][/tex]
[tex]\[ 2a = 0 \][/tex]
[tex]\[ a = 0 \][/tex]
Next, substituting [tex]\( a = 0 \)[/tex] into the equation [tex]\( a + b = -1 \)[/tex]:
[tex]\[ 0 + b = -1 \][/tex]
[tex]\[ b = -1 \][/tex]
We have found that:
[tex]\[ a = 0 \][/tex]
[tex]\[ b = -1 \][/tex]
[tex]\[ c = 2 \][/tex]
Thus, the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( x(x^2 - 1) \)[/tex] is:
[tex]\[ 0x^2 - x + 2 = -x + 2 \][/tex]
Therefore, the remainder is:
[tex]\[ -x + 2 \][/tex]
