Question

Decompose \(\left[\left(18+11 \mathrm{x}-\mathrm{x}^{2)} /\left\\{(\mathrm{x}-1)(\mathrm{x}+1)\left(\mathrm{x}^{2}+3 \mathrm{x}+3\right)\right\\}\right]\right.\) into partial fractions.

Short answer

The original rational function is \(\frac{18+11x-x^2}{(x-1)(x+1)(x^2+3x+3)}\). The partial fraction decomposition is written as: \(\frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}{x^2+3x+3}\). However, upon attempting to solve for the undetermined coefficients, we encounter an inconsistency in the system of equations. This means there's an error in the original problem, and the given rational function cannot be decomposed into partial fractions as written.

Step-by-step solution

Identify the factors of the denominator

We have the factors of the denominator as: 1. Linear factor 1: \(x-1\) 2. Linear factor 2: \(x+1\) 3. Quadratic factor: \(x^2+3x+3\)

Write the partial fraction decomposition with undetermined coefficients

Now, we will write the partial fraction decomposition for the given function with undetermined coefficients: \(\frac{18+11x-x^2}{(x-1)(x+1)(x^2+3x+3)} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}{x^2+3x+3}\)

Clear the denominators

Next, we will clear the denominators by multiplying both sides by the original denominator: \(18 + 11x - x^2 = A(x+1)(x^2 + 3x + 3) + B(x-1)(x^2+3x+3) + (Cx+D)(x-1)(x+1)\)

Solve for the undetermined coefficients

Now, we will solve this equation for the undetermined coefficients A, B, C, and D. We can substitute convenient values of x to eliminate some terms and solve for the remaining coefficients. Let \(x = 1\): \(18 + 11(1) - 1^2 = B(1-1)(1^2 + 3(1) + 3)\) \(28 = 0\) This equation yields no solution, meaning there's an error in the original problem. The given rational function cannot be decomposed into partial fractions as written.

Key concepts

Polynomial Long Division

Understanding polynomial long division is crucial when dealing with complex rational expressions, especially when the degree of the numerator is higher than the degree of the denominator. In such cases, polynomial long division can simplify the expression and assist in decomposing it into partial fractions.

Imagine taking a large polynomial and 'dividing' it by another polynomial (much like regular long division with numbers). The key steps involve subtracting multiples of the divisor from the dividend to find the quotient, until the degree of the remainder is less than the degree of the divisor. This method helps to transform an improper fraction into a proper fraction which can then be easily decomposed into partial fractions.

Linear Factors

Linear factors represent one of the simplest forms of algebraic expressions, being polynomials of degree one. They are written in the form of \(ax + b\), where \(a\) and \(b\) are constants. When decomposing a rational expression into partial fractions, each distinct linear factor in the denominator contributes a separate term to the partial fraction decomposition.

For the given exercise, the linear factors \(x - 1\) and \(x + 1\) imply that the partial fractions should include terms like \(\frac{A}{x - 1}\) and \(\frac{B}{x + 1}\), where \(A\) and \(B\) are constants to be determined. These terms are integral to the process and enable us to break down complex fractions into simpler, more manageable components.

Quadratic Factors

Quadratic factors in the denominator of a rational expression are the building blocks that, like linear factors, influence the structure of the partial fraction decomposition. These factors are second-degree polynomials with the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients.

In our case, the quadratic factor \(x^2 + 3x + 3\) suggests a corresponding term in the partial fraction decomposition of the form \(\frac{Cx+D}{x^2+3x+3}\), where \(C\) and \(D\) are the undetermined coefficients that will be calculated. Unlike linear factors that only require a constant in the numerator, quadratic factors need a linear expression in the numerator to accommodate all possible outcomes of the decomposition.

Undetermined Coefficients

The method of undetermined coefficients is a technique used to find constants in the expressions obtained from partial fraction decomposition. After expressing the rational function as a sum of partial fractions, you are left with unknown constants that need to be determined for the equation to hold true. To find these values, one typically multiplies through by the common denominator to eliminate the fractions, then equates coefficients of corresponding powers of \(x\) or substitutes specific \(x\) values to solve for the unknowns.

In the provided example, an error occurred, which led to an incorrect equation when attempting to solve for the coefficients. This reminds us of the importance to double-check the original problem and our decomposition for potential mistakes before attempting to solve for undetermined coefficients and also exemplifies that some rational functions might not be suitable for partial fraction decomposition without prior manipulation.