Problem 27 - Precalculus Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{x^{2}+1}{x^{3}+x^{2}}$$

Step-by-Step Solution

Answer

\(\frac{x^2 + 1}{x^3 + x^2} = \frac{-1}{x} + \frac{1}{x^2} + \frac{2}{x+1}\).

Step 1: Factor the Denominator

The first step in finding the partial fraction decomposition is to factor the denominator. The given denominator is \(x^3 + x^2\). We can factor out the greatest common factor, which is \(x^2\). Thus, \(x^3 + x^2 = x^2(x+1)\).

Step 2: Set Up Partial Fraction Form

The next step is to set up the partial fraction form based on the factors of the denominator. With the factors \(x^2\) and \(x+1\), the partial fraction decomposition will have the form:\[\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}\]where \(A\), \(B\), and \(C\) are constants to be determined.

Step 3: Clear the Denominator

Multiply each term by the common denominator \(x^2(x+1)\) to eliminate the fractions:\[x^2 + 1 = A(x)(x+1) + B(x+1) + C(x^2)\].

Step 4: Expand and Simplify

Expand the right side of the equation:\[x^2 + 1 = Ax^2 + Ax + Bx + B + Cx^2\]Combine like terms to get:\[x^2 + 1 = (A + C)x^2 + (A + B)x + B\].

Step 5: Equate Coefficients

Equate the coefficients of each power of \(x\) from both sides:1. For \(x^2\): \(1 = A + C\)2. For \(x\): \(0 = A + B\)3. For the constant term: \(1 = B\)

Step 6: Solve the System of Equations

Solve the system of equations:1. From \(1 = B\), we get \(B = 1\).2. Substitute \(B = 1\) into \(A + B = 0\) to find \(A = -1\).3. Substitute \(A = -1\) into \(1 = A + C\) to find \(C = 2\).

Step 7: Write the Partial Fraction Decomposition

Substitute the values of \(A\), \(B\), and \(C\) back into the partial fraction form: \[\frac{x^2 + 1}{x^3 + x^2} = \frac{-1}{x} + \frac{1}{x^2} + \frac{2}{x+1}\].

Key Concepts

Rational Functions

Rational functions are expressions that can be represented as the quotient of two polynomials. In simpler terms, they are fractions where both the numerator and the denominator are polynomials. For example, in the expression \( \frac{x^{2}+1}{x^{3}+x^{2}} \), \(x^2 + 1\) is the numerator and \(x^3 + x^2\) is the denominator.

Understanding rational functions is key because they often appear in calculus and algebra problems. They can be used to represent real-world situations such as rates of change and have properties that allow for the analysis of behavior at specific points.

Some important things to remember about rational functions include:

• The domain, which consists of all real numbers except those where the denominator is zero.
• Asymptotes, which are lines that the graph of the rational function approaches but never touches.
• Roots or zeros of the numerator, which are the values of \(x\) for which the function equals zero.

Understanding these fundamentals will help in grasping concepts like partial fraction decomposition, which is a method used to break down complex rational functions into simpler components.

Factoring

Factoring is the process of breaking down a mathematical expression into a product of simpler terms. For polynomials, this means expressing them as a product of primes, or irreducible polynomials.

In the context of the given problem, factoring plays a crucial role right from the start. The denominator \(x^3 + x^2\) is not in its simplest form, and the step-by-step solution shows that factoring out the greatest common factor \(x^2\) simplifies the denominator to \(x^2(x+1)\).

Here are some tips to keep in mind when factoring:

• Look for a greatest common factor (GCF) among the terms. This can simplify the factoring process significantly.
• Recognize special patterns such as difference of squares, perfect square trinomials, and sum/difference of cubes.
• For quadratic expressions, the FOIL method or other techniques like completing the square can be used.

Effective factoring sets the stage for solving problems that require simplifying expressions or finding roots of polynomials.

System of Equations

A system of equations is a set of equations with multiple variables that are solved together. In the context of partial fraction decomposition, once we have set the partial fraction form, we equate coefficients to form a system of equations.

In our problem, after expanding and equating coefficients, we formed the following system:

• \(1 = A + C\)
• \(0 = A + B\)
• \(1 = B\)

Once this system is established, the solution involves straightforward substitution and elimination methods:

• Assign \(B = 1\) directly from one equation.
• Use substitution into another equation to find \(A = -1\).
• Finally, determine \(C = 2\) by substitution once more.

Utilizing a system of equations enables us to determine the constants \(A\), \(B\), and \(C\) which are essential for writing the partial fraction decomposition.

Polynomials

Polynomials are expressions consisting of variables and coefficients, constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents. They are fundamental building blocks in algebra and form the basis of rational functions.

In polynomials, each term is made up of a base variable raised to an exponent and multiplied by a coefficient. For instance, the polynomial \(x^3 + x^2\) has terms \(x^3\) and \(x^2\).

Understanding the basic properties of polynomials is critical:

• The degree of a polynomial is the highest power of the variable. In \(x^3 + x^2\), the degree is 3.
• Polynomials can be classified by their degree or number of terms (e.g., monomial, binomial, trinomial).
• They can be added, subtracted, multiplied, or divided to create new polynomials.

Recognizing these qualities aids in many mathematical processes, such as factoring polynomials and solving equations, which are integral in finding solutions like the partial fraction decomposition.