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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.

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.

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 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.