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. (7x^2 -9x+3)/(x²+7)²

Verified step by step guidance

1. Step 1: Identify the form of the rational expression. The given expression is \( \frac{7x^2 - 9x + 3}{(x^2 + 7)^2} \).
2. Step 2: Recognize that the denominator \((x^2 + 7)^2\) is a repeated irreducible quadratic factor.
3. Step 3: For a repeated irreducible quadratic factor \((x^2 + 7)^2\), the partial fraction decomposition will include terms for each power of the factor up to the highest power.
4. Step 4: Write the partial fraction decomposition form: \( \frac{Ax + B}{x^2 + 7} + \frac{Cx + D}{(x^2 + 7)^2} \).
5. Step 5: Note that \(A, B, C,\) and \(D\) are constants that would be determined by equating coefficients after multiplying through by the common denominator.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations such as addition, subtraction, multiplication, and division, as well as for decomposing them into simpler components, which is the focus of this question.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational expressions or simplifying complex fractions. The process involves breaking down the expression based on the factors of the denominator, which in this case is (x² + 7)².

Polynomial Degree and Factorization

The degree of a polynomial is the highest power of the variable in the expression. In partial fraction decomposition, understanding the degree helps determine the form of the decomposed fractions. For repeated factors like (x² + 7)², the decomposition will include terms for both the factor itself and its powers, which is essential for correctly setting up the partial fractions.