Problem 2 - Algebra and Trigonometry

Question

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x+7}{(x-1)(x+3)}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational expression is \(A/(x-1) + B/(x+3)\) where A and B are constants.

Step 1: Assign Variables for Constants

Assign variables, say A and B, for the constants in the numerators of the simpler fractions that are to be found.

Step 2: Writing the Form of the Partial Fractions

The expression can be rewritten in the form \(A/(x - 1) + B/(x + 3)\), where A and B are constants.

Key Concepts

Rational Expression

A rational expression is simply a fraction where both the numerator and the denominator are polynomials. The expression \(\frac{5x+7}{(x-1)(x+3)}\) is an example of a rational expression. Here, you have:

• A polynomial in the numerator: \(5x + 7\).
• A polynomial in the denominator: \((x - 1)(x + 3)\).

Rational expressions are similar to regular fractions, but here, both parts contain polynomials. Working with these expressions often involves finding common denominators or simplifying the expressions. In exercises like this one, our main goal is to decompose the expression into simpler parts.

Constants

Constants in mathematical expressions are values that do not change. In the process of partial fraction decomposition, you will often assign constants to unknown values. For the expression \(\frac{5x+7}{(x-1)(x+3)}\), we use constants like \(A\) and \(B\) to stand in for numbers we have not yet calculated.
By assigning these constants, we can express a complex rational expression as a sum of simpler fractions. The beauty of constants is that they provide placeholders, allowing us to work through algebraic manipulations without needing to know their exact values right away.

• In this exercise, the goal is not to find the values of \(A\) and \(B\), but to set up the necessary form for solving.
• This involves assuming \(A\) and \(B\) will have specific values that, when solved, would satisfy the original expression.

Numerators

The numerators are the top parts of fractions. In the partial fraction decomposition of the rational expression, each fraction has its own numerator.
In the exercise provided, the rational expression \(\frac{5x+7}{(x-1)(x+3)}\) is rewritten as two separate fractions: \(\frac{A}{x-1}\) and \(\frac{B}{x+3}\).

• Here, \(A\) and \(B\) are crucial as they become the numerators of each partial fraction.
• The numerators \(A\) and \(B\) help us express how each fraction can add up to reconstruct the original rational expression.

Understanding numerators in this context is important because they ultimately dictate the specific proportions needed to achieve the original expression when combined.

Partial Fractions

Partial fraction decomposition is a method used to break down complex rational expressions into simpler, more manageable pieces called partial fractions. This technique greatly simplifies operations like integration or algebraic manipulation.

• For the rational expression \(\frac{5x+7}{(x-1)(x+3)}\), decomposing it into its partial fractions means writing it as \(\frac{A}{x-1} + \frac{B}{x+3}\).
• The goal of partial fraction decomposition is to make a complicated rational expression easier to handle.

Once decomposed, each simpler fraction can be individually manipulated or solved. It's a powerful technique in calculus and algebra, converting an initially daunting problem into smaller parts that are easier to tackle.