Question

Resolve \(\frac{x}{(1-x)^2}\) into partial fractions.

1. \(\frac{A}{1-x} + \frac{B}{(1-x)^2}\)
2. \(\frac{A}{1-x} + \frac{B}{1+x}\)
3. \(\frac{A}{1+x} + \frac{B}{(1-x)^2}\)
4. \(\frac{A}{1+x} + \frac{B}{1-x}\)

Answer 1

To resolve the fraction \(\frac{x}{(1-x)^2}\) into partial fractions, we can express it as the sum of two fractions: \(\frac{A}{1-x} + \frac{B}{(1-x)^2}\). To find the values of A and B, we can apply the method of finding partial fractions. Solving the system of equations, we find that A = 1 and B = -1.

Step-by-step explanation:

To resolve the fraction

\(\frac{x}{(1-x)^2}\)

into partial fractions, we can express it as the sum of two fractions:

\(\frac{A}{1-x} + \frac{B}{(1-x)^2}\)

To find the values of A and B, we can apply the method of finding partial fractions. First, we multiply both sides of the equation by \((1-x)^2\) to eliminate the denominators:

x = A(1-x) + B

Next, we can simplify and solve for A and B:

x = A - Ax + B

x = (A + B) - Ax

Comparing the coefficients, we get:

A + B = 0

A = 1

Solving the system of equations, we find that A = 1 and B = -1. Therefore, the partial fraction decomposition is:

\(\frac{x}{(1-x)^2} = \frac{1}{1-x} - \frac{1}{(1-x)^2}\)