Problem 58 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. $$\frac{x^{3}-x+3}{x^{2}+x-2}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{x^{3}-x+3}{x^{2}+x-2}\) is \(\frac{6}{x-1}+\frac{1}{x+2}\)

Step 1: Factorize the Denominator

Given \(\frac{x^{3}-x+3}{x^{2}+x-2}\), it is sought to factorize the denominator where \(x^{2}+x-2\). This can be factored as \((x-1)(x+2)\). Therefore, the expression becomes \(\frac{x^{3}-x+3}{(x-1)(x+2)}\).

Step 2: Setting Up Partial Fractions

The next step is to decompose the fraction into two parts based on the factors of the denominator. Write equation as \(\frac{x^{3}-x+3}{x-1} + \frac{x^{3}-x+3}{x+2}\). Set the equation to the original expression \(\frac{x^{3}-x+3}{x^{2}+x-2}\). This results in \(\frac{A}{x-1} + \frac{B}{x+2}\), given that A, B are constants that satisfy the original equation. This is the expansion of \(x^{3}-x+3\) in partial fractions.

Step 3: Finding the Constants A and B

Equating \(\frac{x^{3}-x+3}{x^{2}+x-2}\) and \(\frac{A}{x-1} + \frac{B}{x+2}\) gives us: \(x^{3}-x+3 = A(x+2) + B(x-1)\). By letting x=1, the right-hand side becomes 0 for the \(A\) term, allowing easy calculation of \(B\). Similarly, by letting x=-2, we can find \(A\). Solve the following equation with respect to A and B.

Step 4: Solving the Equation for A and B

After substituting the appropriate values of x into the equation, we find that A = 6 and B = 1. Thus, the decomposed form of the given fraction is \(\frac{6}{x-1}+\frac{1}{x+2}\)

Key Concepts

Rational Expressions

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. These expressions can often become complex, but understanding their structure helps in simplifying and analyzing them.
To fully grasp rational expressions, remember a few key aspects:

• The numerator can be any polynomial.
• The denominator must not be zero, as division by zero is undefined.
• Simplification often involves factoring or using algebraic techniques to break down the expression.

Though they look like a typical fraction, rational expressions represent the ratio of two polynomials. They can be used in a variety of mathematical applications, including solving equations and modeling real-world scenarios. The process of working with these expressions often begins with simplification, as seen in partial fraction decomposition, where the goal is to express a complex rational expression in simpler terms.

Algebraic Fractions

Algebraic fractions differ from numerical fractions in that they involve algebraic expressions—such as variables—in the numerator, the denominator, or both. Understanding algebraic fractions is crucial in simplifying and solving equations involving rational expressions.
Some important points to note when working with algebraic fractions include:

• Simplifying algebraic fractions often requires factoring polynomials in the numerator and denominator.
• We often use methods like cross multiplication to solve equations involving algebraic fractions.
• Partial fraction decomposition is one strategy used to simplify and work further with these fractions.

Simplifying involves making the expression smaller by canceling common factors in the numerator and denominator or by using algebraic identities. The key to mastering algebraic fractions is practice, as this builds confidence in manipulating these expressions through various algebraic techniques.

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that involves breaking down a polynomial into simpler components that, when multiplied together, give the original polynomial. This is especially important in the simplification of rational expressions.
The process of factoring involves several distinct steps:

• Identify and pull out the greatest common factor (GCF), which simplifies the polynomial.
• Use different factoring techniques depending on the form of the polynomial, such as factoring by grouping, using special products (like differences of squares), or applying the quadratic formula.
• Recognize the patterns that simplify the polynomial into binomials or other polynomials.

Having a good grasp of polynomial factoring facilitates not only the simplification of algebraic expressions but also the resolution of algebraic equations. By converting polynomial expressions into a product of simpler expressions, we manage to handle complex algebraic problems more efficiently.