Problem 11 - Precalculus with Limits

Question

Writing the Form of the Decomposition. Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{9}{x^{3}-7 x^{2}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given function is \(\frac{9}{x^{3}-7x^{2}} = \frac{A}{x^{2}} + \frac{B}{x-7}\), where A and B are constants.

Step 1: Factor the Denominator

The denominator of the fraction is \(x^{3}-7 x^{2}\). This expression can be factored. Factoring out an \('x^{2}'\) yields \(x^{2}(x-7)\).

Step 2: Express the Rational Function as a Sum of Partial Fractions

Now take the rational function \(\frac{9}{x^{3}-7 x^{2}}\) and express it as a sum of partial fractions. We have a cubic polynomial in the denominator, which has been factored into \(x^{2}\) and \((x-7)\), yielding two partial fractions. We will denote the constants as A and B (no need to solve for them), so our decomposition is in the form: \(\frac{9}{x^{3}-7x^{2}} = \frac{A}{x^{2}} + \frac{B}{x-7}\)

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is essential when working with partial fraction decomposition. It involves breaking down a polynomial into simpler parts, or 'factors', that when multiplied together give back the original polynomial. This is similar to finding what numbers can be multiplied to obtain another number, like recognizing that 6 can be broken down into 2 and 3.

For example, if we have a quadratic polynomial like \(ax^2 + bx + c\), we try to express it as \( (dx + e)(fx + g) \). Factoring becomes more complex with higher-degree polynomials, as in the textbook problem where \(x^3 - 7x^2\) is factored into \(x^2(x - 7)\). Factors can be numbers, variables, or expressions. In partial fraction decomposition, factoring the denominator is the first crucial step because it defines the structure of the resulting partial fractions.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. The expression \(\frac{9}{x^3 - 7x^2}\) is a rational expression with a constant numerator and a polynomial denominator. Simplifying these expressions often involves factoring, as seen in the decomposition process.

When working with rational expressions, particularly in partial fraction decomposition, the goal is to rewrite them as simpler fractions that are easier to integrate or differentiate, if you are dealing with calculus, or simply easier to understand and calculate with. These simpler fractions have polynomials of lower degrees in the denominator, which are the factors found in the initial factoring step.

Sum of Partial Fractions

The sum of partial fractions is the result of breaking down a complex rational expression into simpler fractions that can be added together to give the original expression. This is particularly handy when integrating rational functions. After factoring the denominator, each factor represents a piece of the partial fraction puzzle.

In the given problem, \(x^3 - 7x^2\) was factored to \(x^2(x - 7)\), yielding two factors: \(x^2\) and \(x-7\). Therefore, the sum of partial fractions will be composed of these pieces: \(\frac{A}{x^2} + \frac{B}{x-7}\), where A and B are constants that would be determined if solving the decomposition completely. These partial fractions make up the building blocks that, when combined, recreate the original rational expression.