Problem 40 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{b^{2}+1}{b^{2}-1} \cdot \frac{b-1}{b+1} $$

Step-by-Step Solution

Answer

\(\frac{b^2+1}{(b+1)^2}\).

Step 1: Identify Factoring Opportunities

We begin by looking for factoring opportunities in the expression. Notice that the denominator of the first fraction, \(b^2-1\), can be factored using the difference of squares identity: \(b^2 - 1 = (b-1)(b+1)\). This will help in simplifying the expression.

Step 2: Rewrite the Expression

Factor \(b^2-1\) in the first fraction to rewrite the expression:\[\frac{b^{2}+1}{(b-1)(b+1)} \cdot \frac{b-1}{b+1}.\]

Step 3: Simplify by Multiplying

Multiply the fractions by multiplying the numerators and the denominators:\[\frac{(b^2+1)(b-1)}{((b-1)(b+1))(b+1)}.\]

Step 4: Eliminate Common Factors

Cancel the common factor \(b-1\) from the numerator and the \(b-1\) term in the denominator, and \(b+1\) (present in the factor) from both numerator and denominator:\[\frac{b^2+1}{(b+1)^2}.\]

Step 5: Final Simplified Form

The expression is now simplified to:\[\frac{b^2+1}{(b+1)^2}.\] There are no further simplifications possible since no common factors remain.

Key Concepts

Factoring in Algebra

Factoring in algebra is an essential skill for simplifying expressions and solving equations. It involves breaking down complex algebraic expressions into a product of simpler factors. Factors are expressions that, when multiplied together, reconstruct the original expression.

• One common method is factoring the difference of squares, where square terms are separated by a subtraction operation, such as in the expression \(b^2 - 1\), which can be simplified using the formula \(a^2 - b^2 = (a-b)(a+b)\).
• Identifying common factors across terms is crucial. Spotting these can make expressions easier to handle and solve.

In this exercise, the term \(b^2 - 1\) in the denominator was recognized as a difference of squares. It is then rewritten as \((b-1)(b+1)\) to simplify the expression. Identifying and applying these formulas make complex algebraic expressions more manageable.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying rational expressions involves making them as simple as possible by factoring and reducing.

• Both the numerator and the denominator should be considered for common factors. If there are any, they can be cancelled, resulting in a simpler fraction.
• Simplifying rational expressions can involve both algebraic manipulations and some arithmetic skills, including recognizing common patterns like factoring opportunities.

In the exercise, simplifying involved factoring \(b^2 - 1\) to identify the common terms. Then, by multiplying across fractions, we can cancel similar terms between the numerator and denominator, reducing the expression to its simplest form. Understanding this concept is crucial for working with more complicated algebraic operations and expressions.

Difference of Squares

The difference of squares is an algebraic technique used to simplify expressions or solve equations. It applies when two squared terms are separated by a subtraction sign. The form is \(a^2 - b^2\) and it always factors into \((a - b)(a + b)\).

• This method is a special case of factoring polynomials that allows for quick simplification.
• The recognition of difference of squares is powerful when simplifying rational expressions or finding roots in equations.

In the exercise, \(b^2 - 1\) is a difference of squares, identified and factored into \((b - 1)(b + 1)\). Recognizing such patterns can make algebraic manipulations quicker and more efficient, sharpening your problem-solving skills in algebra.