Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{5 x^{2}-4}{x^{2}(x+2)} $$

Step-by-Step Solution

Answer

The decomposition is \(\frac{1}{x} - \frac{2}{x^2} + \frac{4}{x+2}\).

Step 1: Identify the Decomposition Form

First, express the given function in a form suitable for partial fraction decomposition. Since the denominator is \(x^2(x+2)\), which is a product of a quadratic term \(x^2\) and a linear term \(x+2\), the decomposition will be of the form: \[\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+2}\]where \(A\), \(B\), and \(C\) are constants to be determined.

Step 2: Multiply through by the Denominator

Clear the fractions by multiplying both sides by the common denominator \(x^2(x+2)\):\[5x^2 - 4 = A x(x+2) + B(x+2) + C x^2\]

Step 3: Expand and Simplify the Right Side

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

Step 4: Set up Equations from Coefficients

Match the coefficients from the expanded form with the corresponding coefficients from the left side \(5x^2 - 4\), giving us:1. \(A + C = 5\)2. \(2A + B = 0\)3. \(2B = -4\)

Step 5: Solve the System of Equations

Solve the system of equations derived in Step 4:- From 3, \(B = -2\)- Substitute \(B = -2\) in 2: \(2A - 2 = 0\) gives \(A = 1\)- Substitute \(A = 1\) in 1: \(1 + C = 5\) yields \(C = 4\)

Step 6: Write the Decomposition

Using the values found for \(A\), \(B\), and \(C\), write the partial fraction decomposition:\[\frac{5x^2 - 4}{x^2(x+2)} = \frac{1}{x} - \frac{2}{x^2} + \frac{4}{x+2}\]

Key Concepts

Rational Expressions

Rational expressions are similar to fractions in that they represent the ratio of two quantities. However, instead of numbers, rational expressions encompass polynomials in the numerator and denominator. They take the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not equal to zero. Rational expressions appear frequently in algebra and higher-level mathematics. They require careful manipulation to be simplified, added, subtracted, multiplied, or divided just like regular fractions.

To work with rational expressions, it's essential to find a common denominator for arithmetic operations, factor polynomials to simplify, and perform polynomial long division where necessary. Partial fraction decomposition, used in the given problem, is an advanced technique for breaking down complex rational expressions into simpler ones. This makes them easier to integrate, differentiate, or analyze further.

Algebraic Fractions

Algebraic fractions, a concept closely related to rational expressions, involve fractions where the numerator and the denominator are algebraic expressions. These expressions include numbers, variables, and the operation symbols such as addition, subtraction, multiplication, and division. Algebraic fractions often require simplification just like numerical fractions, by canceling out common factors between the numerator and the denominator.

In partial fraction decomposition, an algebraic fraction is expressed as a sum or difference of simpler fractions. For example, in the exercise given, the algebraic fraction \(\frac{5 x^{2}-4}{x^{2}(x+2)}\) was decomposed into \(\frac{1}{x} - \frac{2}{x^2} + \frac{4}{x+2}\).

• Each term now is simpler and easier to work with for further calculation.
• This technique is particularly useful in integration, where a complicated integral can be rewritten as a sum of simpler integrals.

Polynomials

Polynomials form the backbone of rational expressions and algebraic fractions. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, \(x^2 + 5x + 6\) is a polynomial in the variable \(x\). The degree of a polynomial is the highest power of the variable present.

In the context of partial fraction decomposition, understanding how to manipulate polynomials is crucial. They can be factored into simpler expressions or divided to simplify rational expressions.

• Factoring helps in identifying the form a rational expression should take for decomposition.
• Recognizing linear and quadratic factors is key to setting up correct partial fractions.

In our exercise, the polynomial \(x^2(x+2)\) in the denominator guided us in choosing the correct form of partial decomposition.

System of Equations

Solving partial fraction decompositions requires setting up and solving a system of equations. This comes into play when determining unknown constants in the decomposition format. After expressing the algebraic fraction in its decomposed form, we multiply through by the denominator to eliminate fractions, which leads to a new equation.

From this new form, we equate like terms --- terms with the same degree of variable --- between the original and decomposed forms, leading to a system of equations.

• For the given problem, three equations were derived: \(A + C = 5\), \(2A + B = 0\), and \(2B = -4\).
• Solving these simultaneous equations gives the constants \(A\), \(B\), and \(C\), which are used to rewrite the original expression.

Mastery over solving system of equations is vital, as it brings precision and correctness in final decompositions.