Problem 33 - College Algebra Essentials

Question

write the partial fraction decomposition of each rational expression. $$ \frac{x+4}{x^{2}\left(x^{2}+4\right)} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the form of the function

The function \( \frac{x+4}{x^{2}(x^{2}+4)} \) is of the form \( \frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator and \( D(x) \) is the denominator. The denominator is a product of two factors \( x^{2} \) and \( x^{2}+4 \), which are not common roots and are degree 2 polynomials. Therefore the expression can be written in the form of: \( \frac{A}{x^2} + \frac{Bx + C}{x^2 + 4} \)

Step 2: Equate to the original expression

The next step is to equate this form with the original expression to solve for A, B, and C. This will involve use of polynomial identities to get rid of the denominators, which results in: \(x+4= A(x^2+4) + x(Bx+ C) = Ax^2 + 4A + Bx^2 + Cx\)

Step 3: Compare coefficients

To find the values of A, B and C, compare the coefficients on both sides. From comparing constants we get 4=4A, so A=1. From comparing the coefficients of x we get that B+C=1. And from comparing the coefficients of \( x^2 \), we find A+B=0, which together with B+C=1, gives B= -1, and C= 2.

Step 4: Write the partial fraction decomposition

Substitute A, B, and C into the general form from Step 1, the partial fraction decomposition of \( \frac{x+4}{x^{2}(x^{2}+4)} \) will be \( \frac{1}{x^2} - \frac{x - 2}{x^2 + 4} \).

Key Concepts

Rational Expression

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. For example, the expression \( \frac{x+4}{x^{2}(x^{2}+4)} \) is a rational expression. The numerator here is a simple linear polynomial, \( x + 4 \), while the denominator is a product of two polynomials, \( x^2 \) and \( x^2 + 4 \).

These expressions are quite common in algebra and calculus. Key properties include:

• The denominator must not be equal to zero. If it is, the expression becomes undefined.
• Simplification can sometimes reveal removable discontinuities or make calculus operations easier.

Knowing how to decompose such expressions into simpler partial fractions is crucial for integration and certain types of algebraic manipulation.

Polynomial Identities

Polynomial identities are equations that hold true for all values of the variable in the expression. They provide a powerful way to manipulate polynomials within rational expressions.

For partial fraction decomposition, we employ polynomial identities to express a complex rational expression as a sum of simpler fractions. This process involves equating a rational expression to the sum of fractions and using the identity to compare terms directly. This can help break down a more complex expression into easier components for further analysis or integration.

Coefficient Comparison

Coefficient comparison is a method used to determine unknown coefficients when expanding or transforming polynomials. Once the rational expression is rewritten in the form of partial fractions, we can apply coefficient comparison.

In our original solution, to find the constants \( A, B, \) and \( C \), we rewritten the expression \( x+4 = A(x^2+4) + x(Bx+C) \).

• Comparing constant terms gave \( 4 = 4A \) leading to \( A = 1 \).
• The linear terms in \( x \) gave us \( B + C = 1 \).
• Comparing the quadratic terms gave \( A + B = 0 \) which resolved to \( B = -1 \) and \( C = 2 \).

By carefully comparing coefficients of the same degree terms, we can solve for unknowns, simplifying the entire expression.

Algebra Techniques

Algebra is full of techniques that enable us to transform and simplify expressions. Partial fraction decomposition utilises a blend of these methods to break down complex rational expressions.

• Identifying and factoring the polynomial in the denominator is crucial for writing down the partial fractions.
• Once identified, expressing the rational expression in terms of unknown coefficients is key. This process involves rewriting the function and expanding it to match the original.
• The algebraic manipulation includes expanding polynomials, gathering like terms, and systematically comparing coefficients, as seen in previous steps.

These techniques not only aid in simplifying algebraic expressions but also enable their integration, a vital process in higher mathematics.