Question

In Problems 17-50, find the partial fraction decomposition of each rational expression. $$ \frac{4}{x(x-1)} $$

Short answer

\(\frac{4}{x(x-1)} = \frac{-4}{x} + \frac{4}{(x-1)} \)

Step-by-step solution

- Factor the Denominator

First, identify and factor the denominator into simpler terms. In this case, the denominator is already factored as \( x(x - 1) \).

- Set Up Partial Fraction Form

Write the partial fraction decomposition as the sum of fractions with unknown coefficients. For \( \frac{4}{x(x-1)} \), we write it as: \[ \frac{4}{x(x-1)} = \frac{A}{x} + \frac{B}{(x-1)} \]

- Clear the Denominator

Multiply both sides of the equation by the common denominator \( x(x - 1) \): \[ 4 = A(x - 1) + Bx \]

- Expand and Combine Like Terms

Expand the right-hand side of the equation: \[ 4 = Ax - A + Bx \] Combine like terms: \[ 4 = (A + B)x - A \]

- Solve for Coefficients

Set up equations by comparing coefficients on both sides: For the constant term: \[ -A = 4 \] For the \(x\)-term: \[ A + B = 0 \] Solve the system of equations: \[ A = -4 \] Substitute \(A\) into \(A + B = 0\): \[ -4 + B = 0 \] \[ B = 4 \]

- Write the Final Decomposition

Substitute the values of \(A\) and \(B\) back into the partial fractions: \[ \frac{4}{x(x-1)} = \frac{-4}{x} + \frac{4}{(x-1)} \]

Key concepts

rational expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. These expressions can sometimes seem complex, but breaking them down can make them more manageable. Rational expressions are often simplified by factoring. Factoring helps us to identify common factors and cancel them out, simplifying the expression. For example, in the given exercise, the rational expression is \(\frac{4}{x(x-1)}\), which is already presented in a factored form. The goal of partial fraction decomposition, as seen in this example, is to express a complicated rational expression as a sum of simpler fractions, making it easier to integrate or differentiate.

factoring denominators

Factoring is an essential skill in algebra. It involves breaking down a complex polynomial into products of simpler polynomials. This step is crucial when working with rational expressions. In our example with \(\frac{4}{x(x-1)}\), the denominator is already factored as \(x(x-1)\). If the denominator wasn't factored, the first step in finding the partial fraction decomposition would be to factor it. Factoring helps in identifying terms that can be decomposed into simpler fractions. When we factor denominators, we look for patterns like quadratic trinomials, difference of squares, and other factorizable polynomials. By factoring, we pave the way to further simplify and solve polynomial equations.

solving systems of equations

Solving systems of equations is a key step in partial fraction decomposition. After setting up the decomposition, you will often have equations with unknown coefficients. You need to solve these equations to find the values of the coefficients. In the exercise, after clearing the denominator and simplifying, we have: \[4 = A(x - 1) + Bx\], which simplifies to \[4 = (A + B)x - A\]. We then compare coefficients to form a system of equations: \[-A = 4\] and \[A + B = 0\]. Solving these equations, we get \[A = -4\] and \[B = 4\]. This process involves setting up and solving linear systems, a fundamental aspect of algebra. Solving systems can be done using methods like substitution, elimination, or matrix operations, all of which help us find the unknown variables in polynomial equations.