Expand the quotients in Exercises 1–8 by partial fractions.
(t⁴ + 9) / (t⁴ + 9t²)

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First, factor the denominator \(t^4 + 9t^2\) by taking out the common factor \(t^2\), so it becomes \(t^2(t^2 + 9)\).

Set up the partial fraction decomposition for the expression \(\frac{t^4 + 9}{t^2(t^2 + 9)}\). Since \(t^2\) is a repeated linear factor, and \(t^2 + 9\) is an irreducible quadratic, write it as: \(\frac{A}{t} + \frac{B}{t^2} + \frac{Ct + D}{t^2 + 9}\).

Multiply both sides of the equation by the denominator \(t^2(t^2 + 9)\) to clear the fractions, resulting in: \(t^4 + 9 = A t (t^2 + 9) + B (t^2 + 9) + (Ct + D) t^2\).

Expand the right-hand side and collect like terms in powers of \(t\) to form a polynomial equation: \(t^4 + 9 = A t^3 + 9 A t + B t^2 + 9 B + C t^3 + D t^2\).

Equate the coefficients of corresponding powers of \(t\) on both sides to form a system of equations, then solve for \(A\), \(B\), \(C\), and \(D\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions, making integration or simplification easier. It involves factoring the denominator and writing the original fraction as a sum of fractions with those factors as denominators.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into products of simpler polynomials. For partial fractions, factoring the denominator completely into linear or irreducible quadratic factors is essential to set up the correct form of the decomposition.

Handling Repeated and Quadratic Factors

When the denominator contains repeated factors or irreducible quadratic factors, the partial fraction setup must include terms for each power of the repeated factor and linear numerators for quadratic factors. This ensures the decomposition accounts for all components of the denominator.

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