Problem 9 - Precalculus: Mathematics for Calculus

Question

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{x^{3}-4 x^{2}+2}{\left(x^{2}+1\right)\left(x^{2}+2\right)}$$

Step-by-Step Solution

Answer

Partial fractions: \(\frac{Ax+B}{x^2+1} + \frac{Cx+D}{x^2+2}\).

Step 1: Identify the Denominator

The denominator of the given rational function is \((x^2 + 1)(x^2 + 2)\). This is already factored into irreducible quadratics.

Step 2: Write Partial Fraction Form

For each irreducible quadratic factor in the denominator, \(x^2 + 1\) and \(x^2 + 2\), set up partial fractions. A partial fraction with an irreducible quadratic denominator has the form \(\frac{Ax + B}{x^2 + c}\). Therefore, the partial fraction form for the function is: \[ \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{x^2 + 2} \] where \(A, B, C,\) and \(D\) are constants that need to be determined.

Key Concepts

Irreducible Quadratics

In mathematics, especially in algebra, quadratics are equations of the form \(ax^2 + bx + c\). Sometimes, these quadratics cannot be factored into real numbers. That means they have no real roots that you can find using standard methods such as factoring. These are known as **irreducible quadratics**. For example, consider \(x^2 + 1\). There are no two real numbers whose product is 1 and sum is 0; thus it cannot be broken down further using real numbers. The concept of irreducibility is important in partial fraction decomposition because it tells us that we will have a more complex form of partial fractions, like \(\frac{Ax + B}{x^2 + c}\), to account for these factors. This is what allows us to analyze complex rational functions by breaking them into simpler parts. Especially in calculus and higher-level algebra, understanding this concept makes it much easier to integrate or differentiate complex expressions.

Rational Functions

A rational function is any function which can be expressed as the fraction of two polynomials, that is \(\frac{P(x)}{Q(x)}\), where both \(P\) and \(Q\) are polynomials and \(Q(x) eq 0\). In our particular exercise, we had \(\frac{x^3 - 4 x^2 + 2}{x^2 + 1) (x^2 + 2)}\). Rational functions can be quite complex because the behavior of these functions can vary greatly depending on the specific form of the polynomials involved. However, we can simplify the process of working with them by expressing them as partial fractions. This decomposes a complex rational function into simpler, more manageable pieces that are easier to integrate or differentiate. By understanding rational functions, you'll be able to approach problems involving polynomials with more confidence.

Coefficients

In any partial fraction decomposition, the coefficients are the constants that need to be determined in order to break down the expression fully. In the context of the rational function from our example, these coefficients are \(A, B, C,\) and \(D\). They represent the unknown values that, once calculated, will split the function into its decomposed form. Calculating these coefficients usually involves setting up equations based on the numerator relationships and then solving for these unknowns using methods like substitution or equating coefficients. While in our exercise, we didn't evaluate these coeffcients, knowing how to find them is crucial when you need to reach the numerical answer, such as when solving integrations. Think of coefficients as pieces of a puzzle, which, when correctly identified, give us a complete picture of the function's decomposition.