Problem 45 - Algebra and Trigonometry

Question

For each expression, determine whether it is already a partial fraction decomposition, or whether it can be decomposed further. (a) \(\frac{x}{x^{2}+1}+\frac{1}{x+1} \quad\) (b) \(\frac{x}{(x+1)^{2}}\) (c) \(\frac{1}{x+1}+\frac{2}{(x+1)^{2}} \quad\) (d) \(\frac{x+2}{\left(x^{2}+1\right)^{2}}\)

Step-by-Step Solution

Answer

(a) decomposed, (b) further decomposable, (c) decomposed, (d) further decomposable.

Step 1: Understanding (a)

Examine \( \frac{x}{x^{2}+1} + \frac{1}{x+1} \). The expression is a sum of two fractions with distinct denominators. Each term resembles a partial fraction form. The denominator \(x^2 + 1\) cannot be factored into linear factors over the real numbers, so \(\frac{x}{x^{2}+1}\) is already a partial fraction. The fraction \(\frac{1}{x+1}\) also has a linear denominator, indicating it's already in a decomposed form.

Step 2: Aliquots for (b)

Consider \( \frac{x}{(x+1)^{2}} \). The denominator is a single irreducible quadratic term \((x+1)^{2}\), which means that if it were to be decomposed further, it would take the form \( \frac{A}{x+1} + \frac{B}{(x+1)^{2}} \). However, it is a single term right now, and not decomposed yet.

Step 3: Assessing (c)

Look at \( \frac{1}{x+1} + \frac{2}{(x+1)^{2}} \). Both terms have denominators that are powers of \(x+1\), and they follow partial fraction notation. The given expression explicitly shows separate terms with the correct reduction, meaning it is already in a partial fraction decomposition.

Step 4: Investigating (d)

Examine \( \frac{x+2}{\left(x^{2}+1\right)^{2}} \). The denominator is a repeated irreducible quadratic term \((x^{2} + 1)^{2}\), and thus can be decomposed further. It can be broken up into a form involving \(\frac{Ax+B}{x^2+1}\) and \(\frac{Cx+D}{(x^2+1)^2}\), where each component represents constancy conditions in terms of powers of \(x^2+1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can represent a wide range of mathematical situations. In the context of partial fraction decomposition, we deal with expressions forming the numerator of a rational function or polynomial fractions.

These expressions can be as simple as polynomials like \(x^3 + 2x^2 + 1\), or more complex, such as the ones involved in partial fraction decomposition. The goal in problems such as these is to simplify complex expressions into sums of simpler fractions. This relates heavily to algebraic manipulations and operations.

• Understand that coefficients represent the number in front of the variable.
• Think of variables as placeholders that can have various values. For instance, in \(3x^2 + 2x + 1\), "x" represents a variable.

Algebraic expressions can often appear in different forms in partial fraction decomposition, held together by operations like addition or subtraction. Mastering these can ease understanding more complex rational expressions.

Rational Functions

Rational functions are vital parts of algebra and calculus, consisting of ratios of polynomials. In our context, these expressions take the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials.

Understanding rational functions helps one grasp how partial fraction decomposition simplifies fractions with polynomial denominators into a series of simpler, easily manageable fractions. This is specifically helpful in integration, where complex rational functions are broken down into partial fractions.

• The numerator \(P(x)\) is generally of lower degree than the denominator \(Q(x)\).
• Ensure that the leading coefficient of the numerator isn't greater than that of the denominator if aiming to use partial fractions.

Recognizing rational functions enables students to break them down into more approachable components by identifying polynomial characteristics in both the numerator and denominator.

Quadratic Factors

Quadratic factors are critical in the decomposition process, particularly when your polynomial expression includes terms that cannot be simplified into linear factors. An irreducible quadratic factor is one that cannot be factored into real linear components. For example, in expressions like \(x^2 + 1\), which remains irreducible over the reals.

Quadratic factors in partial fraction decomposition require specific attention as they need different coefficients. When an expression includes squares of such factors, further levels of complexity need to be handled carefully.

• Quadratic factors often involve terms like \(ax^2 + bx + c\).
• An irreducible factor means expressions like \(x^2 + 1\), which cannot be factored further.

In decomposition, identifying these factors accurately ensures that every distinct polynomial form is addressed correctly, making simplification for integration or solving more straightforward.