Question

Resolve \(\frac{x+1}{x^{2}-4}\) into partial fractions. (1) \(\frac{3}{2(x-2)}+\frac{1}{2(x+2)}\) (2) \(\frac{3}{2(x-2)}-\frac{1}{2(x+2)}\) (3) \(\frac{3}{4(x-2)}+\frac{1}{4(x+2)}\) (4) \(\frac{3}{4(x-2)}-\frac{1}{4(x+2)}\)

Short answer

Question: Determine which of the following options is correct for the partial fraction decomposition of \(\frac{x+1}{x^{2}-4}\). Options: 1. \(\frac{1}{2(x-2)} + \frac{1}{2(x+2)}\) 2. \(\frac{1}{4(x-2)} + \frac{3}{4(x+2)}\) 3. \(\frac{3}{4(x-2)} + \frac{1}{4(x+2)}\) 4. \(\frac{1}{4(x-2)} - \frac{3}{4(x+2)}\) Answer: 3

Step-by-step solution

Factor the denominator

Factor the denominator of the given fraction, which is a difference of two squares: $$x^2 - 4 = (x-2)(x+2)$$

Set up the equation with constants A and B

Now we set up the equation with constants A and B: $$\frac{x+1}{(x-2)(x+2)} = \frac{A}{x-2} + \frac{B}{x+2}$$

Clear the denominators

To eliminate the denominators, multiply both sides by the original denominator \((x-2)(x+2)\): $$x+1 = A(x+2) + B(x-2)$$

Solve for A and B

Now we have a linear equation with A and B as the unknowns. Substitute \(x=2\) to find A: $$2+1 = A(2+2) + B(2-2)$$ $$3 = 4A$$ $$A = \frac{3}{4}$$ Substitute \(x=-2\) to find B: $$-2+1 = A(-2+2) + B(-2-2)$$ $$-1 = -4B$$ $$B = \frac{1}{4}$$

Write the partial fraction expression

Now that we have found A and B, construct the partial fraction expression: $$\frac{x+1}{x^{2}-4} = \frac{\frac{3}{4}}{x-2} + \frac{\frac{1}{4}}{x+2}$$ $$\frac{x+1}{x^{2}-4} = \frac{3}{4(x-2)} + \frac{1}{4(x+2)}$$ The correct expression is option (3).

Key concepts

Algebraic Fractions

Algebraic fractions are a crucial part of algebra. They behave much like numerical fractions but use algebraic expressions. With algebraic fractions, you divide one polynomial by another. This exercise teaches how to handle fractions with polynomial denominators.When working with algebraic fractions, simplifying or decomposing them helps solve problems, especially in calculus and other higher mathematics. One way to simplify an algebraic fraction is through partial fraction decomposition. For instance, in our problem, the fraction \(\frac{x+1}{x^2-4}\) is decomposed to understand it better.Key points to remember:

• Factoring is essential: Break the denominator into its simplest factors.
• Partial fractions: A method where a complex fraction is expressed as the sum of simpler fractions.

This allows easier integration and solving of equations and improves understanding of the relationship between different expressed algebraic fractions.

Polynomial Equations

Polynomial equations play a significant role in breaking down algebraic fractions. A polynomial equation involves variables with whole number exponents. The degree of the polynomial equation is the highest power of the variable present.In our problem, the denominator \(x^2-4\) is a polynomial. It fits the form \(a^2-b^2\), also known as the difference of squares:\[x^2 - 4 = (x - 2)(x + 2)\]This fact changes the fraction's structure, enabling partial fraction decomposition.Important considerations:

• Recognize polynomial identities like the difference of squares.
• Factor polynomial equations as necessary. Here, \(x^2 - 4\) splits into \((x-2)(x+2)\).

Understanding these allows easier manipulation of algebraic fractions and connects the concept to formulating and solving complex equations.

Mathematical Problem Solving

Mathematical problem solving requires a structured approach. The key is breaking down the problem into smaller, manageable parts. Solving equations using partial fractions provides insight into the logical step-by-step approaches used in algebra.Using partial fraction decomposition involves a consistent process:1. **Factor:** Begin by factoring the denominator completely.2. **Set up the system:** Assign constants to the decomposed fractions.3. **Eliminate the denominator:** Multiply through by the original denominator, revealing manageable equations.4. **Solve:** Identify each constant by plugging in strategic values for \(x\) to simplify finding each constant individually.Effective problem solving requires patience and practice:

• Follow each step methodically to avoid mistakes.
• Checking each solution as you progress helps ensure accuracy.

Approaching mathematical problems in a structured, step-by-step method enhances understanding and success.