Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x^{4}}{(x-1)^{3}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \( \frac{x^{4}}{(x-1)^{3}} \) is \( \frac{A}{(x - 1)} + \frac{B}{(x - 1)^{2}} + \frac{C}{(x - 1)^{3}} \). Coefficients A, B, C are determined by solving a system of equations obtained from equating coefficients in the polynomial identity. The decomposition is verified algebraically by combining fractions to reform the original function, and verified graphically by plotting both the original and decomposed functions, ensuring they overlap.

Step 1: Write the general form of partial fractions

Because the denominator has a repeated linear factor, the decomposition will contain terms of the form \( \frac{A}{(x - 1)} \), \( \frac{B}{(x - 1)^{2}} \), and \( \frac{C}{(x - 1)^{3}} \). Write out the general form: \( \frac{x^{4}}{(x-1)^{3}} = \frac{A}{(x - 1)} + \frac{B}{(x - 1)^{2}} + \frac{C}{(x - 1)^{3}} \)

Step 2: Solve for coefficients A, B, C

Multiply through by the common denominator \((x-1)^{3}\), cancel out terms, and collect like terms to form a polynomial identity. Equate the coefficients of corresponding terms on both sides to form a system of equations, and solve this system to determine the coefficients A, B, and C.

Step 3: Verify algebraically

After determining the coefficients, substitute them back into the general form of partial fractions. Then, combine these fractions together to see if it equals to the original function.

Step 4: Verify graphically

Plot the original function and the decomposed function separately but on the same graph. If the decomposition is correct, the two functions will perfectly overlap with each other.

Key Concepts

Rational Expressions

A rational expression is much like a fraction, but instead of integers or real numbers, it has polynomials in the numerator and denominator. Think of it as a fraction that contains algebraic terms. These expressions are used extensively in algebra, often requiring simplification or manipulation to solve equations.

To simplify a rational expression, you may factorize the polynomials and cancel out common factors. However, not all rational expressions can be simplified easily, and that's where partial fraction decomposition comes in handy. It's particularly useful when integrating rational functions in calculus or when working with Laplace transforms in engineering.

When you come across a complex rational expression, especially one with a higher-degree polynomial in the numerator than in the denominator, or with repeating factors in the denominator, partial fraction decomposition enables you to break it down into simpler terms that are easier to work with.

Polynomial Identity

In algebra, a polynomial identity is an equation that is true for all values of the variables within polynomials. For instance, the factoring formula \(a^2 - b^2 = (a+b)(a-b)\) is an identity because it holds true for all numbers a and b.

In the context of partial fraction decomposition, once you've set up the decomposition with variables representing unknown coefficients, you multiply through by the common denominator to eliminate the fractions. The result is an identity where each side of the equation has to match for all values of x. Since the two polynomials are identical, you know that the coefficients of the corresponding powers of x must also be identical. This key principle allows us to solve for the unknown coefficients in the partial fractions.

Repeated Linear Factors

When dealing with partial fraction decomposition, repeated linear factors pose a special scenario. These factors are present when a term in the denominator of a rational expression is raised to a power, such as \( (x - 1)^3 \) in the given exercise.

For each repeated factor, you need to include a corresponding number of terms in your partial fraction decomposition. Each term will have the same linear factor but raised to a different power, starting from 1 and increasing to the power of the repeated factor. This ensures that when you ultimately combine these partial fractions, you account for the repetition of the factor in the original rational expression.

To solve for the coefficients of these terms, you'll employ polynomial identity and equate coefficients, as with any standard partial fraction decomposition. But remember, the presence of repeated factors often results in a more complex system of equations to solve.

Graphing Rational Functions

Graphing rational functions can be quite challenging because of their complexity and the variety of shapes they can take on. They can have vertical asymptotes, horizontal asymptotes, and even oblique asymptotes, depending on the relative degrees of the numerator and denominator polynomials.

Graphing is also an excellent way to verify the accuracy of partial fraction decomposition. By graphing both the original rational function and the decomposed fractions on the same axes, we can visually check if they overlap entirely. They should align perfectly if the decomposition has been done correctly. Additionally, graphing helps us understand the behavior of the function better, such as identifying asymptotes and intercepts, and observing the intervals of increase and decrease.