Question

Resolve \(\frac{\mathrm{x}^{2}+\mathrm{x}+1}{\mathrm{x}^{3}+1}\) into partial fractions. (1) \(\frac{1}{3(x+1)}+\frac{x+1}{3\left(x^{2}-x+1\right)}\) (2) \(\frac{-1}{3(x+1)}+\frac{2 x+2}{3\left(x^{2}-x+1\right)}\) (3) \(\frac{1}{3(x+1)}+\frac{2 x+2}{3\left(x^{2}-x+1\right)}\) (4) None of these

Short answer

Question: Decompose the expression \(\frac{x^2 + x + 1}{x^3 + 1}\) into partial fractions. Answer: \(\frac{1}{3(x+1)}+\frac{2 x+2}{3\left(x^{2}-x+1\right)}\)

Step-by-step solution

Factorize the denominator.

Factorize the denominator x^3 + 1, rewrite it as x^3 +1 = (x+1)(x^2 - x +1). Now the original expression is: \(\frac{x^2 + x + 1}{(x+1)(x^2 - x + 1)}\).

Find the partial fraction decomposition of the expression.

We need to find constants A, B and C such that: \(\frac{x^2 + x + 1}{(x+1)(x^2 - x + 1)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 - x + 1}\). To find these constants, we clear the denominators by multiplying both sides of the equation by (x+1)(x^2 - x + 1): \(x^2 + x + 1 = A(x^2 - x + 1) + (Bx + C)(x+1)\).

Expand and combine like terms.

Expand and combine like terms: \(x^2 + x + 1 = A(x^2 - x + 1) + (Bx^2 + Bx + Cx + C)\). \(x^2 + x + 1 = Ax^2 - Ax + A + Bx^2 + Bx + Cx + C\). \(x^2 + x + 1 = (A + B)x^2 + (-A + B + C)x + (A + C)\).

Equate the coefficients of like terms and solve for A, B and C.

Equate coefficients of like terms: A + B = 1, -A + B + C = 1, A + C = 1. Solving this system of simultaneous equations, we find A = 1/3, B = 2/3 and C = 2/3.

Write the partial fraction decomposition.

Substitute the values of A, B, and C back into the partial fraction decomposition: \(\frac{x^2 + x + 1}{(x+1)(x^2 - x + 1)} = \frac{1/3}{x+1} + \frac{(2/3)x + (2/3)}{x^2 -x + 1}\). Now, the expression has been decomposed into its partial fractions, and the correct answer is: \(\frac{1}{3(x+1)}+\frac{2 x+2}{3\left(x^{2}-x+1\right)}\) (option 3).

Key concepts

Polynomial Factorization

Factoring polynomials is a pivotal skill in algebra that involves expressing a polynomial as the product of its simpler polynomial factors. In our case, we started with the denominator of the expression given in the exercise: \[x^3 + 1.\]We can recognize this as a sum of cubes, which has a standardized factoring approach. A sum of cubes, such as \[a^3 + b^3,\]can be factored into \[(a + b)(a^2 - ab + b^2).\]Applying this to the expression yields:\[x^3 + 1 = (x + 1)(x^2 - x + 1).\]This factorization is crucial, as it simplifies the expression and sets the stage for the partial fraction decomposition. Essentially, we reduce a complex problem into more manageable parts, which is a fundamental aspect of solving many algebraic problems. This approach is not only recommended but necessary for working with rational expressions in partial fractions.

System of Simultaneous Equations

Once we factor the denominator, the next step is to express the given rational expression as a sum of simpler fractions. We achieve this in part by setting up a system of equations based on the coefficients of a polynomial.To decompose the expression \[\frac{x^2 + x + 1}{(x+1)(x^2 - x + 1)},\]we assume it can be written as:\[\frac{A}{x+1} + \frac{Bx + C}{x^2 - x + 1}.\]To find the values of \(A\), \(B\), and \(C\), we clear the denominators by multiplying through by \[(x+1)(x^2 - x + 1),\]giving us:\[x^2 + x + 1 = A(x^2 - x + 1) + (Bx + C)(x+1).\]After expanding and collecting like terms, we equate coefficients of corresponding powers of \(x\) on both sides. This results in a system of simultaneous equations:

• \(A + B = 1\)
• \(-A + B + C = 1\)
• \(A + C = 1\)

These equations can be solved using substitution or elimination methods, yielding the values \(A = \frac{1}{3}\), \(B = \frac{2}{3}\), and \(C = \frac{2}{3}\). Solving such systems allows us to find the constants required for the partial fraction decomposition.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. When working with these expressions, especially in calculus and algebra, simplifying them through partial fraction decomposition can prove tremendously beneficial.In this exercise, our original rational expression was:\[\frac{x^2 + x + 1}{x^3 + 1}.\]By factoring the denominator, we transformed it into a form that suited decomposition. The goal of partial fraction decomposition is to express this rational expression as the sum of simpler fractions whose denominators are the polynomial factors.The technique is particularly useful when dealing with integrals, as simpler expressions are often easier to integrate. Moreover, by aligning with partial fraction decomposition, we prepare a foundation for further mathematical operations such as integration and Laplace Transforms.This method provides more insight into the structure of the expression, allowing for hands-on manipulation in solving more complex algebraic problems.