Problem 47 - Precalculus with Limits

Question

Improper Rational Expression Decomposition, write the partial fraction decomposition of the improper rational expression. $$\frac{x^{4}}{(x-1)^{3}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the improper rational expression \(\frac{x^{4}}{(x-1)^{3}}\) is \(x + 3 + \frac{1}{x - 1} - \frac{2}{(x - 1)^{2}} + \frac{1}{(x - 1)^{3}}\).

Step 1: Identify the decomposition form

Because the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, we first need to do polynomial division. The general form of the decomposition of an expression with the form \((x-a)^n\) in the denominator is like this: \(\frac{A}{x−a} + \frac{B}{(x−a)^2} + \frac{C}{(x−a)^3} + ...\), where A, B, C are constants to be found, and the number of terms depends on the power 'n' of the denominator.

Step 2: Perform Polynomial Division

Since the degree of the numerator is higher than the denominator, we perform polynomial division. Divide the polynomial \(x^{4}\) by \( (x-1)^{3}\). The result is \(x + 3\). So, we can express \(\frac{x^{4}}{(x-1)^{3}} = x + 3 + \frac{D}{x - 1} + \frac{E}{(x - 1)^{2}} + \frac{F}{(x - 1)^{3}}\).

Step 3: Equate Coefficients

Simplify the right hand side of the equation to the same form as the left hand side. Multiply everything out, and combine like terms. We know by comparing coefficients that \(D+E+F = 0\), \(D+E = -3\), and \(D = 1\). We now have three equations with three unknowns.

Step 4: Solve System Of Equations For Constants

Solve this system of equations to find the constants D, E, and F. The result will be \(D = 1, E = -2, F = 1\).

Key Concepts

Improper Rational Expression

An improper rational expression occurs when the degree (the highest power of x) of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. Here, in the expression \(\frac{x^{4}}{(x-1)^{3}}\), the numerator has a degree of 4 while the denominator has a degree of 3, making it improper.
This designation is crucial because it dictates that we need to perform certain additional steps, like polynomial division, before finding the partial fraction decomposition. By understanding whether an expression is proper or improper, you can decide the correct approach to break it down into simpler parts.

Polynomial Division

Polynomial division is a method similar to long division, but it is used when dealing with polynomials instead of numbers.
In the case of the expression \(\frac{x^{4}}{(x-1)^{3}}\), we need to divide the polynomial in the numerator, \(x^{4}\), by the polynomial in the denominator, \((x-1)^{3}\). The result of this division is \(x + 3\).
After performing this division, the remaining part needs to be decomposed into simpler fractions with respect to the denominator. The result \(x + 3\) becomes part of the solution while the remainder is further simplified using partial fractions.

Coefficients

In the context of partial fractions, coefficients are the constants that appear in front of the variables in the expansion. In our example, after performing polynomial division, the remaining fraction is \(\frac{D}{x - 1} + \frac{E}{(x - 1)^{2}} + \frac{F}{(x - 1)^{3}}\).
These coefficients \(D\), \(E\), and \(F\) are unknowns that need to be determined. By simplifying and comparing both sides of the equation to have the same structure, we equate terms with the same degree of the variable, allowing us to derive values for these coefficients.

System of Equations

A system of equations is a collection of equations that you solve together to find a common solution.
Here, after equating coefficients, we identify the following system of equations:
\(D + E + F = 0\), \(D + E = -3\), and \(D = 1\).
These equations must be solved simultaneously to determine the values of \(D\), \(E\), and \(F\). Solving this system reveals that \(D = 1\), \(E = -2\), and \(F = 1\).
Mastering systems of equations is key, as it opens the door to finding the necessary coefficients in rational expressions and many other mathematical problems.