Problem 8 - 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. $$\frac{2 \sqrt{11}}{x^{2}-11}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \( \frac{1}{x - \sqrt{11}} - \frac{1}{x + \sqrt{11}} \).

Step 1: Identify the Expression

The given rational expression is \( \frac{2\sqrt{11}}{x^2 - 11} \). We need to focus on the denominator \( x^2 - 11 \).

Step 2: Factor the Denominator

The denominator \( x^2 - 11 \) is already in its simplest form. It represents a difference of squares, which can be factored as \( (x - \sqrt{11})(x + \sqrt{11}) \).

Step 3: Set Up the Partial Fraction Decomposition

Since the denominator is factored as \( (x - \sqrt{11})(x + \sqrt{11}) \), the partial fraction decomposition is set as:\[ \frac{2\sqrt{11}}{x^2 - 11} = \frac{A}{x - \sqrt{11}} + \frac{B}{x + \sqrt{11}} \]

Step 4: Clear the Denominator

Multiply through by the common denominator \((x - \sqrt{11})(x + \sqrt{11})\) to eliminate fractions:\[ 2\sqrt{11} = A(x + \sqrt{11}) + B(x - \sqrt{11}) \]

Step 5: Expand and Combine Like Terms

Expand the equation:\[ 2\sqrt{11} = Ax + A\sqrt{11} + Bx - B\sqrt{11} \]Combining like terms gives:\[ 2\sqrt{11} = (A + B)x + (A - B)\sqrt{11} \]

Step 6: Equate Coefficients and Solve for Constants

Since there is no \(x\) term in \(2\sqrt{11}\), equate the coefficients of \(x\):\[ A + B = 0 \].Equate the constant term coefficients:\[ A\sqrt{11} - B\sqrt{11} = 2\sqrt{11} \] which simplifies to:\[ A - B = 2 \].Now solve these linear equations:1. \( A + B = 0 \) implies \( A = -B \).2. Substitute \( A = -B \) into \( A - B = 2 \) to get \( -B - B = 2 \), which simplifies to \( -2B = 2 \) or \( B = -1 \).3. Substitute \( B = -1 \) into \( A = -B \) to find \( A = 1 \).

Step 7: Write the Final Partial Fraction Decomposition

Substituting the values of \(A\) and \(B\) into the partial fraction form gives:\[ \frac{2\sqrt{11}}{x^2 - 11} = \frac{1}{x - \sqrt{11}} - \frac{1}{x + \sqrt{11}} \]

Key Concepts

Rational Expressions

Rational expressions are like fractions but with polynomials in the numerator and denominator. They show us how quantities can be divided. Just as with numeric fractions, we aim to simplify rational expressions by factoring both the numerator and the denominator wherever possible. This simplification can make them easier to work with, especially when adding, subtracting, or decomposing them into partial fractions. In our exercise, the given rational expression is \( \frac{2\sqrt{11}}{x^2 - 11} \). Here, our numerator is simply \( 2\sqrt{11} \), and the denominator is a polynomial \( x^2 - 11 \), representing a key part we need to factor. Rational expressions are often analyzed by breaking down their components, allowing us to better understand and manipulate them.

Difference of Squares

The difference of squares is a standout technique in algebra for factoring special polynomials. It applies when we have a form like \( a^2 - b^2 \). This expression can be reframed as \((a - b)(a + b)\), thanks to its inherent structure. In this exercise, \( x^2 - 11 \) might not initially look like a "classic" difference of squares because 11 is not a perfect square. However, if we consider \( 11 \) as \( (\sqrt{11})^2 \), it becomes clear that we can express \( x^2 - 11 \) in this form. Hence, it factors into \((x - \sqrt{11})(x + \sqrt{11})\). This allows us to decompose complex rational expressions into simpler parts, using the foundation of the difference of squares.

Factoring

Factoring is the process of breaking down an expression into multiple simpler elements, or 'factors,' that, when multiplied together, produce the original expression. It is a vital skill in algebra because it helps simplify polynomial expressions and is crucial for partial fraction decomposition. When we take \( x^2 - 11 \), the 'factoring' process reimagines it as \((x - \sqrt{11})(x + \sqrt{11})\). Factoring here is crucial because it paves the way for the partial fraction decomposition, which requires the expression to be expressed in terms of these simpler components to separate them into individual fractions.

Coefficients

Coefficients are the numerical or constant components that multiply variables within an expression or equation. In our example, coefficients help us match and simplify expressions by equating terms with similar powers.In Step 6 of the solution, when dispersing the equation \(2\sqrt{11} = (A + B)x + (A - B)\sqrt{11}\), comparing the 'terms' allows us to determine values for \(A\) and \(B\). Specifically:

• Since there are no \(x\) terms in \(2\sqrt{11}\), the terms involving \(x\) must add to zero: \(A + B = 0\).
• The term \(2\sqrt{11}\) matches with \((A - B)\sqrt{11}\), allowing us to equate \(A - B = 2\).

By solving these equations, the coefficients \(A = 1\) and \(B = -1\) are determined, finalizing the structure of the decomposed rational expression.