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 will need to find a common denominator for the numerator and denominator of the given equation. The common denominator in this case is x^3 + x.

Multiplying both sides of the equation by x^3 + x, we have:

(x^3 + x) * (x-1 / x³+x) = (x^3 + x) * (A / x + Bx + C / x^2+1)

Simplifying the left side of the equation:

(x-1) = A(x^2 + 1) + Bx(x^2 + 1) + C(x^3 + x)

Expanding and simplifying the right side of the equation:

x - 1 = Ax^2 + A + Bx^3 + Bx + Cx^3 + Cx

Combining like terms on the right side:

0 = (B + C)x^3 + Ax^2 + (B + C)x + A - 1

Comparing the coefficients of the terms on both sides of the equation, we can determine the values of A, B, and C:

B + C = 0 (coefficient of x^3)
A = 0 (coefficient of x^2)
B + C = 1 (coefficient of x)
A - 1 = 0 (constant term)

From the first and third equation, we have B + C = 0 and B + C = 1. These two equations contradict each other, so there is no solution for A, B, and C that satisfies the given equation.

Answer 2

You can always separate into partial fractions!

(x-1) / (x^3+x) = -1/x + (x+1)/(x^2+1)
So A = -1, B = 1, C = 1

Answer 3

Apologies for the mistake in my previous response.

To find the values of A, B, and C, we can separate the given fraction into partial fractions:

(x-1) / (x^3+x) = A / x + B(x+1) / (x^2+1)

Multiplying both sides of the equation by (x^3 + x), we have:

(x-1) = A(x^2 + 1) + B(x^3 + x + x^2 + 1)

Expanding and combining like terms, we get:

x - 1 = (B)x^3 + (A + B)x^2 + (B + 1)x + A

Comparing the coefficients of like powers of x on both sides of the equation, we can determine the values of A, B, and C:

1 = A
1 = B + 1
0 = B

Therefore, A = 1, B = 0, and C = 1.

So the values of A, B, and C are A = 1, B = 0, and C = 1.