Problem 7 - Precalculus

Question

(a) factor the denominator of the given \(\mathrm{ra}\) tional expression; (b) determine the form of the partial fraction decomposition for the given rational expression; and (c) determine the values of the constants in the partial fraction decomposition that you gave in part (b). To help you in spotting errors, use the fact that in part (c), each of the required constants turns out to be an integer. $$\frac{8 x-2 \sqrt{5}}{x^{2}-5}$$

Step-by-Step Solution

Answer

The constants are \(A = 3\) and \(B = 5\); the decomposition is \(\frac{3}{x-\sqrt{5}} + \frac{5}{x+\sqrt{5}}\).

Step 1: Factor the Denominator

The given denominator is \(x^2 - 5\). Notice that this expression can be factored by recognizing it as a difference of squares. Thus, we factor it as \((x - \sqrt{5})(x + \sqrt{5})\).

Step 2: Set Up Partial Fraction Decomposition

Since the denominator \((x - \sqrt{5})(x + \sqrt{5})\) is a product of two linear factors, the partial fraction decomposition can be expressed as \(\frac{A}{x - \sqrt{5}} + \frac{B}{x + \sqrt{5}}\), where \(A\) and \(B\) are constants to be determined.

Step 3: Combine the Partial Fractions

Write the expression with a common denominator: \(\frac{A}{x - \sqrt{5}} + \frac{B}{x + \sqrt{5}} = \frac{A(x + \sqrt{5}) + B(x - \sqrt{5})}{(x - \sqrt{5})(x + \sqrt{5})}\). Simplify the numerator to \((A+B)x + (A\sqrt{5} - B\sqrt{5})\).

Step 4: Equate and Solve for Constants

Equate the numerator of this expression to the numerator of the given rational expression: \(8x - 2\sqrt{5} = (A + B)x + (A \sqrt{5} - B \sqrt{5})\). This gives us the system of equations: 1. \(A + B = 8\) for the \(x\)-terms. 2. \(A \sqrt{5} - B \sqrt{5} = -2\sqrt{5}\) for the constant terms.Solve these equations to find \(A\) and \(B\):- Dividing the second equation by \(\sqrt{5}\), we get \(A - B = -2\).- Add the two equations together to solve for \(A\): \(2A = 6\), so \(A = 3\).- Substitute \(A = 3\) back into the first equation: \(3 + B = 8\), so \(B = 5\).

Step 5: Verify the Solution

Use the values obtained, \(A = 3\) and \(B = 5\), to write the partial fraction decomposition:\(\frac{8x - 2\sqrt{5}}{x^2 - 5} = \frac{3}{x - \sqrt{5}} + \frac{5}{x + \sqrt{5}}\).Ensure each coefficient is correct by substituting back into the expressions for consistency.

Key Concepts

Rational Expressions

Rational expressions are fractions in which both the numerator and the denominator are polynomials. Just like regular fractions, they can be simplified, factored, and decomposed. Understanding how to work with rational expressions is crucial for solving problems involving them.

A rational expression generally looks like \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) eq 0\). This is important because dividing by zero is undefined in mathematics. Rational expressions become increasingly useful in calculus and algebra because they allow us to break down complicated expressions into simpler parts.

The key to working efficiently with rational expressions is mastering partial fraction decomposition, where we express a single, more complex fraction as a sum of simpler fractions. This can make it easier to integrate or differentiate in calculus scenarios.

Factoring Denominators

Factoring the denominator is a critical first step in partial fraction decomposition. To decompose the expression \(\frac{8x - 2\sqrt{5}}{x^2 - 5}\), we first recognize that the denominator is a difference of squares.

A difference of squares has the form \(a^2 - b^2 = (a-b)(a+b)\). In this case, the expression \(x^2 - 5\) can be factored as \((x - \sqrt{5})(x + \sqrt{5})\). This transformation is essential because it gives us the linear factors needed to set up the decomposed form of the expression.

Factoring effectively simplifies complex expressions, leading to solutions for otherwise difficult or seemingly impossible-to-solve rational expressions.

Solving Equations

Once the denominators are factored, the next step is solving the equation you get from partial fraction decomposition.

We start by expressing the rational expression in the form \(\frac{A}{x - \sqrt{5}} + \frac{B}{x + \sqrt{5}} \). To find the constants \(A\) and \(B\), we combine these into a single fraction with the common denominator \((x - \sqrt{5})(x + \sqrt{5})\) and simplify.

We equate this resultant fraction's numerator to the original expression's numerator, which gives the equation:\[8x - 2\sqrt{5} = (A + B)x + (A\sqrt{5} - B\sqrt{5})\]
This leads you to a system of equations based on the coefficients of like terms, which you can solve to find the values of \(A\) and \(B\). In this example, you find \(A = 3\) and \(B = 5\), ensuring each term matches by testing against the original fraction after the values are determined.

Linear Factors

Linear factors, such as \(x - \sqrt{5}\) and \(x + \sqrt{5}\), are fundamental in partial fraction decomposition. Linear factors are polynomials of degree one, making them relatively simple to handle compared to higher-degree polynomials.

In the context of partial fraction decomposition, the goal is to express a complicated rational expression as a sum of simpler fractions. Each of these fractions has a linear factor as its denominator. This structure simplifies the process of integration and differentiation if these steps follow later.

The process includes setting up equations based on these factors, then solving for unknown constants. Understanding linear factors helps students break down and tackle more complex polynomial and rational expression problems step-by-step, building a deeper comprehension and enabling more systematic problem-solving.