91Ó°ÊÓ

Rational expressions are essentially fractions wherein both the numerator and the denominator are polynomials. In the example given, the expression \(\frac{2x^{2}+8x+3}{(x+1)^{3}}\) is a rational expression because the numerator \(2x^{2}+8x+3\) and the denominator \(x+1)^{3}\) are polynomials. Simplifying rational expressions often includes factoring polynomials and cancelling out common factors.

However, when the denominator contains repeated factors, as in the case of \(x+1)^{3}\), we turn to partial fraction decomposition to rewrite the expression as a sum of simpler fractions. This technique not only simplifies complex algebraic fractions but also is very useful in calculus, especially in integrations. One key goal of partial fraction decomposition is to break down the complex rational expression into simpler parts that are easier to work with.

Algebraic fractions, like the one in our exercise, are simply fractions that contain algebraic expressions. Partial fraction decomposition is a method used to break these fractions into a sum of simpler fractions with polynomial denominators of lower degrees.

In essence, we're aiming to represent a complicated fraction as a sum of fractions whose denominators are factors of the original denominator. This is done in the exercise by expressing the given rational expression as \(\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3}\). Each term of this sum is called a 'partial fraction', and A, B, and C are constants that we need to determine. Finding these constants involves creating and solving a system of equations, derived from equating coefficients from both sides of the equation.

A system of equations is a collection of two or more equations with the same set of variables. In the example provided, we are dealing with a system of linear equations generated from the coefficient comparison we initiated in the partial fraction decomposition process.

The step where we set up the equations \[\begin{cases} A = 2 \ 2A + B = 8 \ A + B + C = 3 \ \end{cases}\] arises from the condition that the coefficients of the corresponding \(x^k\) terms on both sides of the original equation must match. Solving a system like this usually involves substitution, elimination, or matrix operations. Once we determine the values for A, B, and C through this process, as in our example \(A = 2, B = 4, C = -3\), we conclude the partial fraction decomposition. Understanding how to manipulate and solve systems of equations is essential when working with partial fraction decomposition, and it has broader applications across algebra and calculus.