Question

Express the rational function as a sum or difference of two simpler rational expressions. \(\frac{1}{x^{2}(x-1)}\)

Short answer

The rational function is \(\frac{-1}{x} + \frac{-1}{x^2} + \frac{1}{x-1}\).

Step-by-step solution

Identify the form of partial fractions

We want to express \(\frac{1}{x^{2}(x-1)}\) as a sum of simpler fractions. Since the denominator has a repeated linear factor \(x^2\) and another distinct linear factor \((x-1)\), the partial fraction decomposition will be: \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1}\).

Set up the equation

Write the equation based on the decomposition: \(\frac{1}{x^{2}(x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1}\). Multiply every term by the denominator \(x^2(x-1)\) to clear the fractions: \[1 = A(x)(x-1) + B(x-1) + C(x)^2\].

Expand and collect like terms

Expand the right side: \[1 = A(x^2 - x) + Bx - B + Cx^2\]. Simplify to get: \(1 = (A + C)x^2 + (B - A)x - B\).

Set coefficients equal

Since the left side is a constant (1), equate the coefficients of like terms from both sides. For \(x^2\), \(A + C = 0\); for \(x\), \(B - A = 0\); and for the constant term, \(-B = 1\).

Solve the system of equations

From \(B - A = 0\), we get \(B = A\). From \(-B = 1\), we derive \(B = -1\), hence \(A = -1\). From \(A + C = 0\), since \(A = -1\), we have \(C = 1\).

Write final expression

Substitute the values of \(A, B,\) and \(C\) back into the partial fractions: \[\frac{1}{x^2(x-1)} = \frac{-1}{x} + \frac{-1}{x^2} + \frac{1}{x-1}\].

Key concepts

Understanding Rational Expressions

Rational expressions are like fractions, but instead of just numbers, they contain polynomials in their numerator and denominator. They form a significant part of algebra as they describe relationships where one quantity is divided by another. A rational expression can look like \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomial functions and \(Q(x) eq 0\).
Similar to numeric fractions, rational expressions need a specific approach for simplification, solving, or operations like addition and subtraction. The focus often lies on finding a common denominator before performing these operations.
In partial fraction decomposition, a complex rational expression is rewritten as the sum of simpler rational expressions. This process is vital for integration and solving differential equations. For instance, simplifying \(\frac{1}{x^2(x-1)}\) using partial fractions helps break it down into simpler fractions that are easier to handle.

Algebraic Fractions and Their Components

Algebraic fractions are a subset of rational expressions where both the numerator and the denominator are polynomials. An example is the expression \(\frac{3x + 5}{x^2 - 4}\).
Just like regular fractions, algebraic fractions can be added, subtracted, multiplied, and divided, with similar rules. However, algebraic fractions involve manipulating polynomial expressions.
The key steps include finding the least common denominator (LCD) for addition or subtraction and factoring the expressions for simplification. Moreover, it's crucial to understand how zeros of the denominator affect the expression by creating asymptotes or undefined points in its graph.
In partial fraction decomposition, algebraic fractions are particularly transformed to show multiple fractions stored within one larger fraction. This helps simplify complex expressions and is often used in calculus for integration purposes. By turning \(\frac{1}{x^2(x-1)}\) into simpler fractions like \(\frac{-1}{x} + \frac{-1}{x^2} + \frac{1}{x-1}\), mathematical operations become more manageable.

Polynomial Long Division for Simplification

Polynomial long division is a method used to divide polynomials, similar to the long division used with numbers. It’s particularly useful when the degree of the numerator is larger than the degree of the denominator. This process is essential in simplifying rational expressions and is sometimes a preliminary step before using partial fraction decomposition.
To perform polynomial long division, you follow these steps:

• Align the dividend under the division bracket.
• Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
• Multiply the entire divisor by this term and subtract it from the dividend.
• Bring down the next term of the dividend and repeat the process until all terms are exhausted.

This method simplifies the expression and makes it easier to decompose further using partial fractions. In the context of the example \(\frac{1}{x^2(x-1)}\), polynomial long division isn’t necessary due to the lower degree of the numerator, but it’s crucial when dealing with more complex expressions with high-degree numerators.