Problem 2 - Algebra 2

Question

Simplify each expression. \(\frac{a+b}{a^{2}-b^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{a-b} \).

Step 1: Identify the Structure

First, recognize that the given expression can be simplified using algebraic identities and factoring. The provided expression is \( \frac{a+b}{a^{2}-b^{2}} \). Notice the denominator \( a^{2}-b^{2} \) is a difference of squares.

Step 2: Apply the Difference of Squares

Rewrite the denominator \( a^{2}-b^{2} \) as a product of two binomials using the identity \( a^{2} - b^{2} = (a-b)(a+b) \). Thus, the expression becomes \( \frac{a+b}{(a-b)(a+b)} \).

Step 3: Simplify the Fraction

The numerator \( a+b \) is the same as one of the factors in the denominator. Thus, you can cancel out \( a+b \) in the numerator and denominator, giving \( \frac{1}{a-b} \). Ensure \( a eq b \) to avoid division by zero.

Key Concepts

Difference of Squares

In algebra, the difference of squares is an essential concept used in simplifying expressions and solving equations. When you have an expression of the form \( a^2 - b^2 \), it is known as a "difference of squares". This is because it represents the gap between two square numbers. The special property of such expressions is that they can be easily factored into a product of two binomials.Here's how the transformation works:

• Recognize that \( a^2 - b^2 \) can be rewritten using the identity: \( (a-b)(a+b) \).
• This identity is based on the geometric basis that the area of a large square minus the area of a smaller square equals the area of two rectangles.

Using this identity, expressions that appear complex can be simplified, which is a powerful tool in algebra. Always look for patterns indicating a difference of squares when simplifying rational expressions.

Factoring Expressions

Factoring is a major technique in algebra that involves breaking down an expression into simpler "factor" components that, when multiplied together, give back the original expression.When faced with an expression, here's how to factor it:

• Look for common factors in each term.
• Apply patterns such as the difference of squares. For example, \( a^2 - b^2 = (a-b)(a+b) \).
• Check if there are special binomial formulas that can apply, like the perfect square trinomial or the sum and difference of cubes.

The ability to factor expressions is vital for simplifying complex equations and for solving quadratic equations, allowing you to break down seemingly complicated problems into more manageable parts.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying these expressions often involves factoring the numerator and the denominator, and then canceling common factors to reduce the expression to its simplest form.Here's a step-by-step guide to simplifying rational expressions:

• Factor both the numerator and the denominator completely.
• Look for and cancel out any common factors, ensuring the denominator is not zero.
• Remember to specify any variable restrictions that arise during cancellation to avoid division by zero.

For example, in the expression \( \frac{a+b}{a^2-b^2} \), you factor the denominator using the difference of squares as \((a-b)(a+b)\). Then, by canceling the common factor of \(a+b\), you simplify the rational expression to \( \frac{1}{a-b} \), as long as \(a eq b\). Mastering the simplification of rational expressions is foundational for higher algebra and calculus.