In Exercises 9–42, write the partial fraction decomposition of each rational expression. 6x²-x+1/(x³ + x²+x+1)

Verified step by step guidance

1. First, factor the denominator $x{x}^3+x^2+x+\; 1$. Group terms to factor by grouping: $(x^3\; +\; x^2)\; +\; (x\; +\; 1)$.
2. Factor out common terms from each group: $x^2(x\; +\; 1)\; +\; 1(x\; +\; 1)$.
3. Since both groups contain $(x\; +\; 1)$, factor it out: $(x\; +\; 1)(x^2\; +\; 1)$.
4. Set up the partial fraction decomposition using the factors of the denominator: $\backslash frac\{6x^2\; -\; x\; +\; 1\}\{(x\; +\; 1)(x^2\; +\; 1)\}\; =\; \backslash frac\{A\}\{x\; +\; 1\}\; +\; \backslash frac\{Bx\; +\; C\}\{x^2\; +\; 1\}$, where $A$, $B$, and $C$ are constants to be determined.
5. Multiply both sides by the denominator $(x\; +\; 1)(x^2\; +\; 1)$ to clear the fractions, then equate coefficients of corresponding powers of $x$ to form a system of equations to solve for $A$, $B$, and $C$.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This technique is especially useful for integrating rational functions or solving equations. It involves breaking down the denominator into factors and writing the original fraction as a sum of fractions with those factors as denominators.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials, called factors. For partial fraction decomposition, factoring the denominator completely is essential because it determines the form of the partial fractions. Techniques include factoring by grouping, synthetic division, or using special formulas.

Degree of Polynomials in Rational Expressions

The degree of a polynomial is the highest power of the variable in the expression. In partial fraction decomposition, the degree of the numerator must be less than the degree of the denominator. If not, polynomial long division is performed first to rewrite the expression as a polynomial plus a proper fraction.