5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

Verified step by step guidance

1. Identify the factors in the denominator: \(x^{2} - 8x + 16\) and \(x^{2} + 3x + 4\). Notice that \(x^{2} - 8x + 16\) is a perfect square trinomial and can be factored as \((x - 4)^{2}\). The other quadratic \(x^{2} + 3x + 4\) does not factor nicely over the reals, so it remains as is.
2. Since \((x - 4)^{2}\) is a repeated linear factor, set up terms for both the first power and the second power of \((x - 4)\) in the decomposition. For each linear factor, the numerator is a constant: \(\frac{A}{x - 4} + \frac{B}{(x - 4)^{2}}\).
3. For the irreducible quadratic factor \(x^{2} + 3x + 4\), the numerator must be a linear expression of the form \(Cx + D\) because the degree of the numerator is one less than the degree of the denominator factor.
4. Combine all parts to write the partial fraction decomposition setup as: \(\frac{A}{x - 4} + \frac{B}{(x - 4)^{2}} + \frac{Cx + D}{x^{2} + 3x + 4}\).
5. This setup is the appropriate form for the partial fraction decomposition of the given expression. The next step (not required here) would be to multiply both sides by the denominator and solve for the constants \(A\), \(B\), \(C\), and \(D\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves breaking down a complex fraction into a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of simpler polynomials. Recognizing perfect square trinomials or irreducible quadratics is essential to determine the correct form of partial fractions, especially when denominators contain quadratic factors.

Form of Partial Fractions for Quadratic Denominators

When the denominator includes irreducible quadratic factors, the corresponding partial fractions have linear numerators (Ax + B) over these quadratics. For repeated or distinct quadratic factors, each term in the decomposition must reflect the degree and multiplicity of the factor.