In Exercises 9–42, write the partial fraction decomposition of each rational expression. (4x²+3x+14)/(x³ - 8)

Verified step by step guidance

1. Recognize that the denominator is a difference of cubes: $x^3-8$. Recall the formula for difference of cubes: $a^3\; -\; b^3\; =\; (a\; -\; b)(a^2\; +\; ab\; +\; b^2)$.
2. Factor the denominator using the difference of cubes formula: $x^3\; -\; 8\; =\; (x\; -\; 2)(x^2\; +\; 2x\; +\; 4)$.
3. Set up the partial fraction decomposition form. Since the denominator factors into a linear factor $(x\; -\; 2)$ and an irreducible quadratic factor $(x^2\; +\; 2x\; +\; 4)$, write the decomposition as: $\backslash frac\{4x^2\; +\; 3x\; +\; 14\}\{(x\; -\; 2)(x^2\; +\; 2x\; +\; 4)\}\; =\; \backslash frac\{A\}\{x\; -\; 2\}\; +\; \backslash frac\{Bx\; +\; C\}\{x^2\; +\; 2x\; +\; 4\}$, where A, B, and C are constants to be determined.
4. Multiply both sides of the equation by the denominator $(x\; -\; 2)(x^2\; +\; 2x\; +\; 4)$ to clear the fractions, resulting in: $4x^2\; +\; 3x\; +\; 14\; =\; A(x^2\; +\; 2x\; +\; 4)\; +\; (Bx\; +\; C)(x\; -\; 2)$.
5. Expand the right-hand side and collect like terms in powers of $x$. Then, equate the coefficients of corresponding powers of $x$ on both sides to form a system of equations. Solve this system to find the values of A, B, and C.

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, typically linear or irreducible quadratic factors.

Factoring the Denominator

Factoring the denominator is essential to identify the simpler components for partial fractions. In this problem, the denominator x^3 - 8 is a difference of cubes, which factors as (x - 2)(x^2 + 2x + 4). Recognizing and applying special factoring formulas helps set up the correct form for decomposition.

Setting Up and Solving for Coefficients

After factoring, assign unknown coefficients to each partial fraction term based on the factors. Multiply both sides by the original denominator to clear fractions, then equate coefficients of corresponding powers of x to form a system of equations. Solving this system yields the values of the unknown coefficients.