Problem 61 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{b^{2}-a^{2}}{a-b} $$

Step-by-Step Solution

Answer

The simplified expression is \(-(b+a)\).

Step 1: Identify the expression

The given expression is \(\frac{b^2 - a^2}{a-b}\). The goal is to simplify this rational expression.

Step 2: Recognize a difference of squares

Notice that the numerator \(b^2 - a^2\) is a difference of squares. It can be factored into \((b-a)(b+a)\).

Step 3: Factor the numerator

Using the difference of squares formula \(x^2-y^2=(x-y)(x+y)\), the expression becomes \(\frac{(b-a)(b+a)}{a-b}\).

Step 4: Rearrange the denominator

Notice that \((a-b) = -(b-a)\), so the expression becomes \(\frac{(b-a)(b+a)}{-(b-a)}\).

Step 5: Cancel out the common term

The \((b-a)\) terms in the numerator and denominator cancel each other out, leaving \(-1(b+a)\) as the simplified expression.

Key Concepts

Algebraic Simplification

Algebraic simplification is a process that helps to make expressions shorter and easier to work with. By performing simplification, we reduce expressions to their simplest form without changing their original value. When simplifying algebraic expressions, the main goal is to identify and eliminate any unnecessary complexity.

For instance, when simplifying a rational expression like \(\frac{b^2-a^2}{a-b}\), we look for patterns or identities that can make the process easier. In this example, recognizing the pattern is crucial for further simplification.

Key steps to remember when simplifying include:

• Identifying common terms
• Recognizing patterns such as the difference of squares
• Reducing fractions by cancelling out common factors

These actions help us work with more manageable forms of the expression and thus offer clarity when solving algebraic problems.

Difference of Squares

The difference of squares is a specific pattern in algebra that involves the subtraction of two squared terms. It takes the form of \(x^2 - y^2\) and is factored using the identity \((x-y)(x+y)\). This identity is crucial in simplifying complex algebraic expressions.

In the given expression \(b^2-a^2\), recognizing it as a difference of squares allows us to apply the formula, simplifying it into \((b-a)(b+a)\). This transformation is what enabled the simplification of the original problem.

Understanding and applying this identity allows you to:

• Efficiently factor quadratic expressions
• Unlock simplifications in equations
• Simplify computations in algebra

Mastering the difference of squares helps tackle more advanced algebraic problems with ease.

Factoring Expressions

Factoring is a fundamental algebraic technique that involves breaking down expressions into their simplest components or factors. When an expression is factored, it becomes a product of factors, which are usually simpler and easier to work with.

For the expression \(b^2-a^2\), factoring involved using the difference of squares to rewrite it as \((b-a)(b+a)\), providing a way to simplify the rational expression by observing that \(b-a\) is a common factor in both the numerator and the denominator.

Steps to successful factoring include:

• Identifying terms and constants that can be factored out
• Using algebraic identities like the difference of squares
• Rewriting equations as a product of simpler expressions

When you practice factoring, you gain techniques for simplifying and solving a wide range of algebraic problems efficiently.