12. What are the values of A, B.C.respectively so that

Question

x-1 / x³+x = A / x + Bx + C / x²+1 ?

Answer 1

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