Problem 56 - College Algebra with Corequisite Support

Question

For the following exercises, find the partial fraction expansion. $$ \frac{x^{3}-4 x^{2}+5 x+4}{(x-2)^{3}} $$

Step-by-Step Solution

Answer

The partial fraction expansion is \(\frac{5}{(x-2)^2} + \frac{14}{(x-2)^3}\).

Step 1: Identify the Form of the Partial Fraction

To find the partial fraction expansion of a rational expression, first, identify the form of the denominator. The denominator is \((x-2)^3\), which is a repeated linear factor.

Step 2: Set Up the Partial Fraction Decomposition

For the repeated linear factor \((x-2)^3\), the partial fraction decomposition will take the form: \[\frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{(x-2)^3}\]where \(A\), \(B\), and \(C\) are constants that can be found.

Step 3: Write the Equation for the Decomposition

The original fraction can be equated to the sum of the partial fractions:\[\frac{x^3-4x^2+5x+4}{(x-2)^3} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{(x-2)^3}\]

Step 4: Clear the Denominators by Multiplying

To clear the denominators, multiply both sides by \((x-2)^3\):\[x^3-4x^2+5x+4 = A(x-2)^2 + B(x-2) + C\]

Step 5: Expand and Simplify the Right Side

Expand the terms on the right side:- \(A(x-2)^2 = A(x^2 - 4x + 4)\)- \(B(x-2) = Bx - 2B\)Combine to get: \[x^3-4x^2+5x+4 = Ax^2 - 4Ax + 4A + Bx - 2B + C\]

Step 6: Equate Coefficients and Solve for Constants

Match coefficients of like terms from both sides of the equation:- For \(x^3\), coefficient is 0, so \(A = 0\)- For \(x^2\), coefficient is -4, so \(-4A = -4\) (already satisfied)- For \(x\), coefficient is 5, so \(-4A + B = 5\)- Constant term is 4, so \(4A - 2B + C = 4\)Using \(A = 0\), simplify to find \(B = 5\) and \(C = 14\).

Step 7: Write the Final Partial Fraction Expansion

Substitute the values found for \(A\), \(B\), and \(C\) back into the partial fraction setup:\[\frac{0}{x-2} + \frac{5}{(x-2)^2} + \frac{14}{(x-2)^3} = \frac{5}{(x-2)^2} + \frac{14}{(x-2)^3}\]

Key Concepts

Rational Expression

A rational expression is like a fraction, but instead of numbers, it involves polynomials. Polynomials are expressions made up of variables raised to powers, often with coefficients in front. In mathematical terms, a rational expression is a ratio of two polynomials. For instance, in the exercise we have the rational expression \( \frac{x^3 - 4x^2 + 5x + 4}{(x-2)^3} \). Here, the numerator is a polynomial of degree 3 and the denominator is a polynomial corresponding to \((x-2)\) raised to the third power.
When solving problems involving rational expressions, especially when trying to simplify or transform them, it's crucial to understand both the numerator and denominator. By breaking them down, you can take complex expressions and turn them into simpler, more manageable pieces. Partial fraction decomposition is a key method to achieve this simplification, enabling easier integration or simplification of equations involving these expressions.

Linear Factor

Linear factors are expressions of the form \(ax + b\) where the degree of the polynomial is 1. In this exercise, the denominator \((x-2)^3\) features a repeated linear factor \((x-2)\). Recognizing this is vital because it sets the pattern for how we decompose or split the rational expression into simpler fractions.
When we have repeated linear factors like in our example \((x-2)^3\), the partial fraction decomposition will require separate terms for each power. That is why our breakdown from the example was: \(\frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{(x-2)^3}\). Each power of the repeated factor gets its own fraction term in the decomposition. This helps us organize and match coefficients more effectively, thereby aiding the solution process.

Coefficient Matching

Coefficient matching is a technique used during partial fraction decomposition to find the unknown constants. Once we multiply through by the denominator to clear the fractions, we expand and reorganize the terms. The goal is to have a polynomial on each side of the equation that looks identical regarding the powers of \(x\). For the example \(x^3-4x^2+5x+4 = Ax^2 - 4Ax + 4A + Bx - 2B + C\), we need matching coefficients for \(x^3, x^2, x,\) and the constant term.
Through this method, each coefficient from one side of the equation is set equal to the corresponding coefficient on the other side. This gives us a system of equations which we solve to find the constants \(A, B, \) and \(C\). In this particular exercise, matching led to simple equations yielding \(A = 0\), \(B = 5\), and \(C = 14\). Coefficient matching allows us to break down what might otherwise be a complex puzzle into a series of straightforward steps.