Write the partial fraction decomposition of each rational expression. 6x²-x+1/(x³ + x² + x +1)

Verified step by step guidance

1. First, identify the rational expression given: \(\frac{6x^{2} - x + 1}{x^{3} + x^{2} + x + 1}\).
2. Next, factor the denominator \(x^{3} + x^{2} + x + 1\). Group terms to factor by grouping: \(x^{2}(x + 1) + 1(x + 1)\).
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. Since \(x + 1\) is linear and \(x^{2} + 1\) is an irreducible quadratic, write: \(\frac{6x^{2} - x + 1}{(x + 1)(x^{2} + 1)} = \frac{A}{x + 1} + \frac{Bx + C}{x^{2} + 1}\), 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, resulting in: \(6x^{2} - x + 1 = A(x^{2} + 1) + (Bx + C)(x + 1)\). Then expand and collect like terms to solve for \(A\), \(B\), 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 with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex rational expressions into manageable parts.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors, which can be linear or quadratic. Factoring the denominator is essential in partial fraction decomposition because it determines the form and number of terms in the decomposition.

Degree of Polynomials

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 division is performed first to rewrite the expression appropriately.