Asked by Taaju
12. What are the values of A, B.C.respectively so that
x-1 / x³+x = A / x + Bx + C / x²+1 ?
x-1 / x³+x = A / x + Bx + C / x²+1 ?
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Answered by
GPT 3.5
To find the values of A, B, and C, we need to simplify both sides of the equation and equate the corresponding coefficients.
Starting with the left side of the equation:
x - 1 / (x^3 + x)
To simplify this expression, we'll need to find a common denominator for the fraction:
x * (x^3 + x) - 1 / (x^3 + x)
= (x^4 + x^2 - 1) / (x^3 + x)
Now let's focus on the right side of the equation:
A / x + Bx + C / (x^2 + 1)
To simplify this expression, we need to find a common denominator for the fraction:
(A * (x^2 + 1) + Bx * x + C) / (x^2 + 1)
= (Ax^2 + A + Bx^2 + C) / (x^2 + 1)
Now we can equate the corresponding coefficients of the numerators:
x^4 + x^2 - 1 = Ax^2 + A + Bx^2 + C
Comparing the coefficients:
For the coefficient of x^4: 1 = 0 (since there is no x^4 term on the right side)
For the coefficient of x^2: 1 = A + B
For the constant term: -1 = A + C
We now have a system of equations:
A + B = 1
A + C = -1
Subtracting the second equation from the first, we get:
(B - C) = 2
We can choose any value for B and let C be 2 less than B. For simplicity, let's choose B = 0. Then C = -2.
Now let's substitute these values into the first equation to find A:
A + B = 1
A + 0 = 1
A = 1
Therefore, the values of A, B, and C are:
A = 1
B = 0
C = -2
Starting with the left side of the equation:
x - 1 / (x^3 + x)
To simplify this expression, we'll need to find a common denominator for the fraction:
x * (x^3 + x) - 1 / (x^3 + x)
= (x^4 + x^2 - 1) / (x^3 + x)
Now let's focus on the right side of the equation:
A / x + Bx + C / (x^2 + 1)
To simplify this expression, we need to find a common denominator for the fraction:
(A * (x^2 + 1) + Bx * x + C) / (x^2 + 1)
= (Ax^2 + A + Bx^2 + C) / (x^2 + 1)
Now we can equate the corresponding coefficients of the numerators:
x^4 + x^2 - 1 = Ax^2 + A + Bx^2 + C
Comparing the coefficients:
For the coefficient of x^4: 1 = 0 (since there is no x^4 term on the right side)
For the coefficient of x^2: 1 = A + B
For the constant term: -1 = A + C
We now have a system of equations:
A + B = 1
A + C = -1
Subtracting the second equation from the first, we get:
(B - C) = 2
We can choose any value for B and let C be 2 less than B. For simplicity, let's choose B = 0. Then C = -2.
Now let's substitute these values into the first equation to find A:
A + B = 1
A + 0 = 1
A = 1
Therefore, the values of A, B, and C are:
A = 1
B = 0
C = -2
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