In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (7x² -9x+3)/(x²+7)²

Verified step by step guidance

1. Identify the denominator and factor it if possible. Here, the denominator is ${x²+7}^2$, which is already factored as $(x^2\; +\; 7)^2$.
2. Since the denominator is a repeated irreducible quadratic factor, the partial fraction decomposition will include terms for each power of this factor up to 2.
3. For each power of the irreducible quadratic factor $(x^2\; +\; 7)$, write a numerator that is a linear polynomial: for the first power, use $Ax\; +\; B$, and for the second power, use $Cx\; +\; D$.
4. Set up the partial fraction decomposition as the sum of these terms: $\backslash frac\{Ax\; +\; B\}\{x^2\; +\; 7\}\; +\; \backslash frac\{Cx\; +\; D\}\{(x^2\; +\; 7)^2\}$.
5. This form represents the partial fraction decomposition of the given rational expression without solving 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 rational expression into a sum of fractions with simpler denominators.

Repeated Quadratic Factors

When the denominator contains a repeated irreducible quadratic factor, such as (x² + 7)², the decomposition includes terms with the quadratic factor raised to increasing powers. Each term has a linear numerator for quadratic factors, reflecting the factor's multiplicity.

Form of Numerators in Partial Fractions

For linear factors, numerators are constants; for irreducible quadratic factors, numerators are linear expressions (ax + b). This ensures the decomposition can represent the original rational function accurately before solving for constants.