Question

Resolve \(\frac{3 x-5}{(x-1)^{4}}\) into partial fractions. (1) \(\frac{1}{(x-1)}+\frac{2}{(x-1)^{2}}-\frac{3}{(x-1)^{3}}+\frac{4}{(x-1)^{4}}\) (2) \(\frac{3}{(x-1)^{2}}+\frac{2}{(x-1)^{3}}-\frac{1}{(x-1)^{4}}\) (3) \(\frac{3}{(x-1)^{3}}+\frac{2}{(x-1)^{4}}\) (4) \(\frac{3}{(x-1)^{3}}-\frac{2}{(x-1)^{4}}\)

Short answer

Answer: The partial fraction decomposition is \(\frac{3}{(x-1)^{3}} - \frac{2}{(x-1)^{4}}\).

Step-by-step solution

Identify the partial fraction decomposition setup

The given fraction is \(\frac{3 x-5}{(x-1)^{4}}\). The denominator has a single factor, \((x-1)\), raised to the power of \(4\). Therefore, the appropriate partial fraction decomposition setup will include four terms, with the powers of \((x-1)\) from \(1\) to \(4\). So, our setup is: \(\frac{3 x-5}{(x-1)^{4}} = \frac{A}{(x-1)} + \frac{B}{(x-1)^{2}} + \frac{C}{(x-1)^{3}} + \frac{D}{(x-1)^{4}}\)

Clear the denominators

To find the constants A, B, C, and D, we need to clear the denominators by multiplying both sides by \((x-1)^{4}\): \((3 x-5) = A(x-1)^{3} + B(x-1)^{2} + C(x-1) + D\)

Solve for the constants

Now, we need to solve for the constants A, B, C, and D by expanding and collecting like terms: First, choose some values of x so that some of the terms become zero: For x = 1: \(-2 = D\) Now, we differentiate both sides of the equation with respect to x: \(3 = 3A(x-1)^{2} + 2B(x-1) + C\) For x = 1: \(3 = 3A(0) + 2B(0) + C \Rightarrow C = 3\) Now, differentiate again: \(0 = 6A(x-1) + 2B\) For x = 1: \(0 = 6A(0) + 2B \Rightarrow B = 0\) Finally, differentiate once more: \(0 = 6A\) \(\Rightarrow A = 0\)

Write the partial fractions

Now that we have the values of the constants A, B, C, and D, we can write the partial fractions: \(\frac{3 x-5}{(x-1)^{4}} = \frac{0}{(x-1)} + \frac{0}{(x-1)^{2}} + \frac{3}{(x-1)^{3}} - \frac{2}{(x-1)^{4}}\) So the partial fraction decomposition is: \(\frac{3}{(x-1)^{3}} - \frac{2}{(x-1)^{4}}\)

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). The expression represents a constant value or a variable that changes depending on its components. For example, \(3x - 5\) is an algebraic expression with a variable x, and constants 3 and -5.

In the context of the exercise given, the expression \(\frac{3x-5}{(x-1)^4}\) represents a rational expression, where the numerator is an algebraic expression and the denominator is a polynomial raised to a power. The challenge students often face is manipulating these expressions to simplify them or, as in this exercise, decompose them into simpler parts or 'partial fractions' which are easier to work with in calculus and other applications.

Understanding algebraic expressions is critical for students because it helps to develop an intuition about how variables interact and how a change in one part of the expression can affect the whole.

Polynomial Factorization

The process of breaking down polynomials into a product of simpler factors is known as polynomial factorization. It's like finding the original pieces of a puzzle that fit together to give you the whole picture. For instance, a quadratic polynomial like \(x^2 - 1\) can be factored into \(x + 1)(x - 1)\).

In our exercise, we deal with the expression \(x-1\), which is already in its factored form. However, it is raised to a power (\(x-1)^4\), indicating repeatability of this factor. Understanding factorization is vital because it helps simplify complex polynomials and allows for techniques such as partial fraction decomposition to be used. This foundational skill aids students in solving equations, finding zeros or roots of the polynomials, and in calculus operations such as integration.

Rational Expressions

A rational expression is the ratio of two polynomials, like a fraction, where both the numerator and the denominator are algebraic expressions themselves. The exercise provided is an exemplar of such an expression: \(\frac{3x-5}{(x-1)^4}\).

Rational expressions can often be simplified, or in cases where we want to integrate or solve equations, decomposed into partial fractions. This is particularly useful when the denominator has multiple factors, which can be factored out. In the given exercise, the denominator is already a single repeated factor raised to the power of 4, meaning the focus is on breaking the expression into a sum of terms with decreasing powers of \(x-1\).

Understanding rational expressions and their manipulation is crucial to working with equations and functions within algebra and calculus. Students must grasp this concept to solve more complex mathematical problems effectively.