Problem 3 - Calculus of a Single Variable

Question

Partial Fraction Decomposition In Exercises \(1-4,\) write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-3}{x^{3}+10 x} $$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational expression is \(\frac{A}{x}\) + \(\frac{Bx + C}{x^{2}+10}\).

Step 1: Factorize the denominator

Start by factorizing the denominator of the rational expression. In this case, \(x^{3}+10x\) can be written as \(x(x^{2}+10)\).

Step 2: Write the form of the partial fraction decomposition

The given expression can now be decomposed into partial fractions. The general form of the partial fractions would be \(\frac{A}{x}\) + \(\frac{Bx + C}{x^{2}+10}\), where A, B, and C are constants to be determined (we are not solving for them in this exercise).

Key Concepts

Factorization

When working with algebraic expressions, especially rational expressions, factorization lays the groundwork for many advanced techniques such as partial fraction decomposition. Factorization is a process used to break down polynomials into simpler, multiplied factors that are easier to work with.

For instance, in the exercise, the polynomial in the denominator, \(x^3 + 10x\), is factored into \(x(x^2 + 10)\). This factorization simplifies the polynomial by revealing its roots. In this context, one root is at \(x=0\), corresponding to the factor \(x\), and the others are complex since \(x^2 + 10\) does not factor further over the real numbers.

Understanding factorization is crucial because it makes the next steps of identifying the partial fraction form possible. Just remember the basic principle: every polynomial can be factored into a product of irreducible polynomials and/or a product of binomials raised to some powers, where irreducible means the polynomial cannot be factored further in the set of real numbers.

Rational Expressions

Rational expressions are fractions wherein the numerator and the denominator are both polynomials. The exercise presented showcases a rational expression: \(\frac{2x-3}{x^3+10x}\). A key aspect of understanding rational expressions lies in how we manage their denominators.

When dealing with rational expressions, always look for opportunities to simplify or transform by factoring, as this can often make the next stages of manipulation or decomposition clearer and more straightforward. The simplification of rational expressions is similar to the simplification of numeric fractions—it involves breaking them down into their simplest form.

For many operations, including solving equations or integrating rational functions in calculus, having the expression in simplified form is critical. Thus, gaining confidence with rational expressions will provide a sturdy foundation as you venture into more complex areas of mathematics.

Algebraic Fractions

Algebraic fractions are just like your regular fractions, but instead of numbers, they contain algebraic expressions in their numerators and denominators. The primary goal when working with algebraic fractions is to make them as uncomplicated as possible. This can be achieved through various methods, including factorization and partial fraction decomposition.

Partial fraction decomposition, as demonstrated in the exercise, is a method for breaking apart complex algebraic fractions into simpler, more manageable pieces. This technique is particularly useful when integrating rational expressions or solving complex algebraic equations. The decomposition seeks to express a complicated fraction as a sum of simpler fractions, where each simpler fraction's denominator is a factor of the original denominator.

Each term in the partial fraction decomposition, like \(\frac{A}{x}\) and \(\frac{Bx + C}{x^2+10}\) in the exercise, corresponds to a partial fraction. These simpler fractions can be dealt with individually, which simplifies the overall process and makes subsequent algebraic manipulation more clear-cut. Mastery of algebraic fractions not only eases the handling of rational expressions but also enriches problem-solving skills across various applications in algebra and calculus.