Problem 33 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition. Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{x}{x^{3}-x^{2}-2 x+2}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the rational expression \(\frac{x}{{x^{3}-x^{2}-2 x+2}}\) is \(\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{x-2}\), where A, B, and C are constants that we have determined. Verification by adding these fractions together will result in the original rational expression.

Step 1: Factorizing the Denominator

First, we factorize the denominator \(x^{3}-x^{2}-2 x+2\) of the function. We can factorize it to \((x-1)(x+1)(x-2)\)

Step 2: Forming the Partial Fractions Decomposition

We know that the partial fraction decomposition of the function will be in the form \(\frac{x}{(x-1)(x+1)(x-2)}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{x-2}\) where A, B, and C are constants that we need to determine.

Step 3: Solving For the Constants

To find the values of A, B, and C, we multiply through by the common denominator to get rid of the fraction, set up equations by comparing coefficients on both sides, and solve the equations.

Step 4: Verifying The Result

After determining the values of A, B, and C, insert them into the partial fraction decomposition and then simplifying. If the result is the same as the original function, then it's correct; otherwise, check the previous steps.

Key Concepts

Factoring Polynomials

Factoring polynomials is a critical step in many areas of algebra, including partial fraction decomposition. It involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give the original polynomial. This is essential because it simplifies complex expressions and allows us to solve polynomial equations more easily.

Consider the polynomial in the denominator of our exercise, \(x^3-x^2-2x+2\). To factor it, one commonly starts by looking for patterns or by using methods such as grouping, synthetic division, or the rational root theorem. In our case, the polynomial factors into \(x-1\), \(x+1\), and \(x-2\).

Mastering factoring helps students to progress through algebra smoothly, and its applications spread across calculus, number theory, and beyond. For example, recognizing that \(x^2-1 = (x-1)(x+1)\) as a difference of squares is an instance of pattern identification leading to easier factorization. The more practice one gets with different types of polynomials, the quicker and more accurate the factoring becomes.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. In algebra, we often encounter complex rational expressions that can be simplified or manipulated for various calculations, making understanding them crucial.

The expression \(\frac{x}{x^{3}-x^{2}-2x+2}\) from our exercise is a rational expression. Simplifying such an expression can lead to easier integration, differentiation, or finding limits in calculus. One of the techniques for simplifying rational expressions is by factoring as seen in the previous section.

Importance of Simplification

By simplifying rational expressions, we can more clearly see their characteristics, such as vertical and horizontal asymptotes, intercepts, and domain. The process often involves factoring polynomials and cancelling common factors between the numerator and the denominator, which reduces the expression to its simplest form.

Algebraic Fractions

Algebraic fractions, also known as fractional algebraic expressions, involve fractions with variables. These are an extension of numerical fractions and follow similar rules for addition, subtraction, multiplication, and division, but they require algebraic manipulation besides arithmetic operations.

In partial fraction decomposition, we break down an algebraic fraction into simpler 'partial' fractions, making them easier to integrate or add. In our exercise, the algebraic fraction \(\frac{x}{(x-1)(x+1)(x-2)}\) is decomposed into partial fractions with unknown constants A, B, and C.

Decomposition and Calculation

Once we have the expressions for the partial fractions, we find the constants by multiplying the entire equation by the common denominator to eliminate the fractions. By setting coefficients of like terms on both sides equal, we create a system of equations that can be solved for A, B, and C, simplifying the task of working with the original complex fraction.