Question

Find the partial fraction decomposition of each rational expression. $$ \frac{2 x+3}{x^{4}-9 x^{2}} $$

Short answer

\[\frac{2x + 3}{x^4 - 9x^2} = \frac{-2/3}{x} + \frac{-1/3}{x^2} + \frac{2/3}{x + 3} + \frac{2/3}{x - 3}\]

Step-by-step solution

Factor the Denominator

First, factor the denominator. The given denominator is \(x^4 - 9x^2\). This can be factored as follows:\[ x^4 - 9x^2 = x^2 (x^2 - 9) = x^2 (x + 3)(x - 3)\] So, the factored form of the denominator is \(x^2 (x + 3)(x - 3)\).

Write the Partial Fraction Decomposition Form

Express the given fraction as a sum of partial fractions. Since the denominator is \(x^2 (x + 3)(x - 3)\), the partial fraction decomposition will have terms involving these factors:\[ \frac{2x + 3}{x^4 - 9x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 3} + \frac{D}{x - 3}\] where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Create a Common Denominator

Multiply each term by the common denominator \(x^2 (x + 3)(x - 3)\) to clear the fractions:\[2x + 3 = A x (x+3)(x-3) + B (x+3)(x-3) + C x^2 (x-3) + D x^2 (x+3)\]

Expand and Collect Like Terms

Expand each term on the right-hand side to simplify:\[2x + 3 = A (x^3 - 3x) + B (x^2 - 9) + C x^3 - 3C x^2 + D x^3 + 3D x^2\]Combine like terms:\[2x + 3 = (A + C + D)x^3 + (-3A - 3C + 3D)x^2 + (-3A + 0B)x + (-9B)\]

Equate Coefficients

Compare the coefficients of like terms on both sides of the equation:For the coefficient of \(x^3\):\[0 = A + C + D\]For the coefficient of \(x^2\):\[0 = -3A - 3C + 3D\]For the coefficient of \(x\):\[2 = -3A\]For the constant term:\[3 = -9B\]

Solve the System of Equations

Solve the system of equations to find the values of \(A\), \(B\), \(C\), and \(D\):From \(2 = -3A\):\[A = -\frac{2}{3}\]From \(3 = -9B\):\[B = -\frac{1}{3}\]Substitute \(A = -\frac{2}{3}\) into the other equations:From \(0 = A + C + D\):\[0 = -\frac{2}{3} + C + D\Rightarrow C + D = \frac{2}{3}\]From \(0 = -3A - 3C + 3D\):\[0 = -3(-\frac{2}{3}) - 3C + 3D\Rightarrow 2 - 3C + 3D = 0\Rightarrow -3C + 3D = -2\Rightarrow D - C = \frac{2}{3}\]Solve for \(C\) and \(D\):Adding the two equations involving \(C\) and \(D\):\[C + D = \frac{2}{3} + D - C = \frac{2}{3}\Rightarrow 2D = \frac{4}{3}\Rightarrow D = \frac{2}{3}\]Therefore, \(C = 0\).

Write the Final Decomposition

Using the values found, write the partial fraction decomposition:\[ \frac{2x + 3}{x^4 - 9x^2} = \frac{-2/3}{x} + \frac{-1/3}{x^2} + \frac{2/3}{x + 3} + \frac{2/3}{x - 3}\]

Key concepts

Factoring Polynomials

Factoring polynomials is a crucial step in partial fraction decomposition. In this scenario, our start point is the polynomial denominator:\[ x^4 - 9x^2 \].Factoring involves breaking down this expression into simpler polynomial factors. Notice that both terms share a common factor of \( x^2 \), allowing us to factor out \( x^2 \):\[ x^4 - 9x^2 = x^2(x^2 - 9) \].The next step involves recognizing the difference of squares in \( x^2 - 9 \), which can be factored further:\[ x^2 - 9 = (x + 3)(x - 3) \].Combining these results, we get the complete factored form:\[ x^4 - 9x^2 = x^2 (x + 3)(x - 3) \].Understanding how to factor polynomials simplifies many algebra problems, including working with rational expressions.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. For our example, the rational expression is:\[ \frac{2x + 3}{x^4 - 9x^2} \].Our task here is to decompose this into simpler fractions. Partial fraction decomposition allows us to express this complex fraction as a sum of simpler ones. After factoring the denominator, our rational expression looks like this:\[ \frac{2x + 3}{x^2(x + 3)(x - 3)} \].We then assume it can be written as:\[ \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 3} + \frac{D}{x - 3} \].Each term now represents a simpler fraction, making the expression easier to handle and integrate. Working with rational expressions requires a good grasp of factoring and basic algebraic manipulation.

Solving Systems of Equations

The partial fraction decomposition process involves determining constants (\( A, B, C, \) and \( D \)), leading us to a system of linear equations. After expressing the given fraction as partial fractions and finding a common denominator, we get:\[ 2x + 3 = A x (x+3)(x-3) + B (x+3)(x-3) + C x^2 (x-3) + D x^2 (x+3) \].Expanding and collecting like terms, we then compare coefficients to form equations like:\[ 0 = A + C + D \]\[ 0 = -3A - 3C + 3D \]\[ 2 = -3A \]\[ 3 = -9B \].These lead to a system of linear equations, which we solve step by step:

• From \( 2 = -3A \): \( A = -\frac{2}{3} \).
• From \( 3 = -9B \): \( B = -\frac{1}{3} \).

By substituting these values back, we solve for \( C \) and \( D \). Systems of equations are essential in algebra for solving multiple unknowns simultaneously.

Algebra

Algebra is the broader mathematical discipline that deals with symbols and the rules for manipulating those symbols. In partial fraction decomposition, we use various fundamental algebraic techniques:

• Factoring polynomials
• Manipulating rational expressions
• Solving systems of equations

Starting from the expression:\[ \frac{2x + 3}{x^4 - 9x^2} \],we employed algebraic rules to factor, decompose, and solve. Each step uses algebraic principles to simplify and solve the problem. This systematic approach is typical in many areas of mathematics, making algebra a versatile and essential tool.