Problem 5 - Precalculus with Calculus Previews

Question

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{3 x^{2}-x+4}{x^{4}+2 x^{3}+x^{2}} $$

Step-by-Step Solution

Answer

The decomposition is \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1} + \frac{D}{(x+1)^2} \).

Step 1: Factor the Denominator

First, we need to factorize the denominator \(x^{4} + 2x^{3} + x^{2}\). Notice here, you can factor out an \(x^{2}\) from each term, giving:\[ x^{4} + 2x^{3} + x^{2} = x^{2}(x^{2} + 2x + 1) \] Further factoring \(x^{2} + 2x + 1\) reveals it as a perfect square:\[ x^{2} + 2x + 1 = (x+1)^{2} \]Thus, the fully factored denominator is \(x^{2}(x+1)^{2}\).

Step 2: Set Up Partial Fractions

Now we express the partial fraction decomposition based on the factors from Step 1. We have terms \(x^{2}\) and \((x+1)^{2}\), leading to the setup:\[ \frac{3x^{2}-x+4}{x^{4}+2x^{3}+x^{2}} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x+1} + \frac{D}{(x+1)^{2}} \]Where \(A\), \(B\), \(C\), and \(D\) are the coefficients to be determined.

Step 3: Formulate the Equation for Decomposition

Multiply through by the common denominator \(x^{4} + 2x^{3} + x^{2}\) to clear fractions:\[ 3x^{2} - x + 4 = A x (x + 1)^{2} + B (x + 1)^{2} + C x^{2} (x + 1) + D x^{2} \]This expression represents the equation that must be satisfied by the numerators wherein the coefficients are yet to be determined.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in the process of partial fraction decomposition. Here, it involves breaking down a polynomial into simpler, non-divisible elements called factors. This is often the first step when working with rational expressions, as it simplifies further operations.

For the given rational expression \( \frac{3x^{2}-x+4}{x^{4}+2x^{3}+x^{2}} \), we initially focus on the denominator. The goal is to express \(x^{4}+2x^{3}+x^{2}\) in its simplest form by identifying common factors.

Notice that each term of the polynomial can have \(x^2\) factored out:

• Rewrite the expression: \(x^{4}+2x^{3}+x^{2} = x^2(x^{2}+2x+1)\).
• Further simplify \(x^{2}+2x+1\) as it is a perfect square: \( (x+1)^2 \).

This results in a fully factored denominator of \(x^2(x+1)^2\). Such simplification allows the application of partial fraction decomposition in the next steps.

Rational Expressions

Rational expressions consist of a numerator and a denominator, both of which are polynomials. Understanding how to manipulate these expressions is vital for solving algebraic problems, especially in calculus.

The decomposition of a rational expression, like \( \frac{3x^{2}-x+4}{x^{4}+2x^{3}+x^{2}} \), entails transforming it into a sum of simpler fractions. After factoring the denominator, we use its factors to set up partial fractions:

• The expression is set up as a sum: \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1} + \frac{D}{(x+1)^2} \).
• Each term in this setup involves undetermined coefficients \(A, B, C, D\).

The purpose of partial fraction decomposition is to handle complex rational expressions by reducing the difficulty of integration or other operations.

Polynomial Division

Polynomial division isn't directly visible in this exercise, but it relates to the understanding and management of rational expressions. Polynomial division could serve as an alternative strategy in simplifying a polynomial when setting up partial fractions doesn't fully satisfy or if dealing with complex expressions.

Essentially, polynomial division can be compared to long division applied in arithmetic but involves polynomials. It is used to divide a larger polynomial (the dividend) by another polynomial (the divisor) to yield a quotient and sometimes a remainder. Although not needed for this specific partial fraction setup, its knowledge is useful when initial simplifications or factorizations do not suffice.
Key steps in polynomial division include:

• Identifying the leading term of the dividend and divisor.
• Determining how many times the leading term of the divisor fits into the leading term of the dividend.
• Subtracting the resulting product from the dividend to form a new, simpler expression.
• Repeating the process until no further division is possible.

Understanding polynomial division can foster a deeper grasp of how polynomials operate, fundamentally supporting the application of partial fraction decomposition.