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. (3x+16)/(x + 1) (x − 2)²

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1. Identify the factors in the denominator: (x + 1) and (x - 2)². Note that (x - 2) is a repeated linear factor with multiplicity 2.
2. Write a separate term for each factor and each power of the repeated factor. For the linear factor (x + 1), use a term of the form $A/(x+1)$ where A is a constant to be determined.
3. For the repeated linear factor (x - 2)², write terms for each power: one term with denominator (x - 2) and another with denominator (x - 2)². These take the form $B/(x-2)$ and $C/(x-2{)}^2$ respectively, where B and C are constants.
4. Combine all terms to write the full partial fraction decomposition: $\frac{A}{x}+\frac{B}{x}+\frac{C}{{(}^x}$.
5. This expression represents the form of the partial fraction decomposition. Constants A, B, and C would be found by multiplying both sides by the denominator and equating coefficients, but that is not required here.

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 quadratic factors.

Factorization of Denominators

Understanding how to factor the denominator into linear and repeated factors is essential. In this problem, the denominator is factored as (x + 1)(x − 2)², indicating one linear factor and one repeated linear factor, which affects the form of the partial fractions.

Form of Partial Fractions for Repeated Factors

When a denominator has repeated linear factors like (x − 2)², the partial fraction decomposition includes terms for each power of the repeated factor. For (x − 2)², the decomposition includes terms with denominators (x − 2) and (x − 2)², each with its own constant numerator.