Problem 5 - Precalculus

Question

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{\text { A polynomial of degree }<9}{(x+4)^{5}\left(x^{2}+1\right)^{2}} $$

Step-by-Step Solution

Answer

Decomposition form: \(\frac{A_1}{x+4} + \frac{A_2}{(x+4)^2} + \frac{A_3}{(x+4)^3} + \frac{A_4}{(x+4)^4} + \frac{A_5}{(x+4)^5} + \frac{Bx+C}{x^2+1} + \frac{Dx+E}{(x^2+1)^2}\)."

Step 1: Identify the Denominator Components

The given denominator is \((x+4)^5\left(x^2+1\right)^2\). Recognize that this is composed of a linear factor \((x+4)\) and a quadratic factor \(\left(x^2+1\right)\).

Step 2: Assign Decomposition Terms for Linear Factor

For the factor \((x+4)^5\), assign terms of the form \(\frac{A_1}{x+4} + \frac{A_2}{(x+4)^2} + \frac{A_3}{(x+4)^3} + \frac{A_4}{(x+4)^4} + \frac{A_5}{(x+4)^5}\).

Step 3: Assign Decomposition Terms for Quadratic Factor

For the quadratic factor \(\left(x^2+1\right)^2\), assign terms of the form \(\frac{Bx + C}{x^2+1} + \frac{Dx + E}{(x^2+1)^2}\).

Step 4: Formulate the Complete Partial Fraction Form

Combine all terms identified in the previous steps to form the partial fraction decomposition basis: \[\frac{A_1}{x+4} + \frac{A_2}{(x+4)^2} + \frac{A_3}{(x+4)^3} + \frac{A_4}{(x+4)^4} + \frac{A_5}{(x+4)^5} + \frac{Bx+C}{x^2+1} + \frac{Dx+E}{(x^2+1)^2}.\]

Key Concepts

Rational Expressions

In algebra, a rational expression is essentially a fraction where both the numerator and the denominator are polynomial expressions. These are quite similar to numeric fractions, but instead of numbers, they involve variables. Rational expressions require careful handling, especially when simplifying or manipulating them, as the presence of variables introduces additional complexity.

If we consider \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, it is crucial that \(Q(x) eq 0\) since division by zero is undefined.

When working with rational expressions, it is often helpful to express them in a simpler form, which can involve factoring, reducing, or decomposing. This is where partial fraction decomposition comes into play, breaking complex rational expressions into simpler, more manageable components.

Polynomial Functions

Polynomial functions form the backbone of rational expressions and are key building blocks in mathematics. A polynomial is an expression consisting of variables, coefficients, and exponents that are non-negative integers. They take the general form: \ a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \. The degree of the polynomial is the largest exponent—which dictates the curve’s shape.

If we have a polynomial of degree less than 9, it implies it might be something like a cubic \(x^3\), quartic \(x^4\), or lower, but not reaching the degree of nine. Polynomial functions can often be factorized, aiding in simplifying and solving rational expressions. Understanding how polynomial functions behave and can be manipulated is essential to handling more complex mathematical equations like in the given exercise.

Denominator Factors

The denominator of a rational expression is especially significant when it comes to partial fraction decomposition. It often contains factors that determine how the rational expression can be broken down. Factors can be linear, such as \(x + 4\), or quadratic, such as \(x^2 + 1\).

In the exercise, the denominator \( (x+4)^5 (x^2+1)^2 \) reveals two distinct types of factors:

• **Linear factors:** These occur when the factor is of the first degree, like \(x + 4\).
• **Quadratic factors:** These arise when the factor is of the second degree, like \(x^2 + 1\).

Understanding these factors is crucial as it determines the form of the decomposition terms. For linear factors raised to a power \(n\), we assign terms like \ \frac{A_n}{(x+4)^n} \—up to the highest power. For quadratic factors, we include terms \ \frac{Bx+C}{x^2+1} \ or \ \frac{Dx+E}{(x^2+1)^2} \ to reflect the polynomial degree in the numerators.