Problem 89 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{6}{b^{2}-9} \cdot \frac{b+3}{2 b+4}$$

Step-by-Step Solution

Answer

\( \frac{3}{(b-3)(b+2)} \)

Step 1: Factor Denominators

First, let's factor the denominator of the first fraction, \( b^2 - 9 \). Notice that this is a difference of squares: \( b^2 - 9 = (b - 3)(b + 3) \).

Step 2: Simplify the Expression

Now rewrite the expression using the factored form: \( \frac{6}{(b-3)(b+3)} \cdot \frac{b+3}{2b+4} \).

Step 3: Cancel Common Factors

In the multiplication, the \( b+3 \) in the numerator of the second fraction cancels with the \( b+3 \) in the denominator of the first fraction.

Step 4: Perform Multiplication

After canceling the common factors, multiply what remains: \( \frac{6}{b-3} \cdot \frac{1}{2b+4} = \frac{6}{(b-3)(2b+4)} \).

Step 5: Simplify the Result

Notice that \( 2b+4 \) can be factored further: \( 2b+4 = 2(b+2) \). Rewrite the expression as \( \frac{6}{2(b-3)(b+2)} \). Then simplify by canceling the common factor of 2 from the numerator and denominator: \( \frac{3}{(b-3)(b+2)} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that allows us to simplify complex expressions. This process involves breaking down a polynomial into the product of its factors, which can often make solving equations and simplifying expressions much easier.
When dealing with polynomials, look for common patterns like the difference of squares. For example, in the expression \( b^2 - 9 \), recognize that this is a difference of squares, written as \((b^2) - (3^2)\). This can be factored into \((b - 3)(b + 3)\).
Other factoring techniques include finding the greatest common factor or applying special formulas for perfect square trinomials or sum and difference of cubes. Mastery of these techniques is crucial for simplifying algebraic expressions and solving equations effectively.

Simplifying Fractions

Simplifying fractions, especially those with algebraic expressions, involves reducing the fraction to its simplest form. This is achieved by canceling out common factors between the numerator and the denominator.
When you have factored forms in the numerator and the denominator, check for any expressions common to both. For example, in the expression \( \frac{6}{(b-3)(b+3)} \cdot \frac{b+3}{2b+4} \), the \(b+3\) terms appear in both a numerator and a denominator. They can be canceled, resulting in a simpler product.
Always look for opportunities to factor further, as seen with \(2b+4\) which simplifies to \(2(b+2)\). This step is crucial for reducing the algebraic fractions fully. It's important to perform this step with care, ensuring that no potential simplifications are overlooked.

Algebraic Operations

Algebraic operations are mathematical procedures used to manipulate expressions and solve equations. These include addition, subtraction, multiplication, and division. When dealing with rational expressions, you apply these operations similarly to regular numbers, but with added steps due to the variables and expressions involved.
In multiplication of rational expressions, you first factor and cancel out any common terms, as seen in the example of multiplying \(\frac{6}{(b-3)(b+3)}\) by \(\frac{b+3}{2b+4}\). After simplifying, perform the multiplication: multiply the remaining numerators together and the denominators together.
Remember to periodically review your work, as each step in the operations can present opportunities to simplify further. The goal is to end with a fully simplified expression like \(\frac{3}{(b-3)(b+2)}\), and understanding these operations can significantly enhance your ability to work with rational expressions overall.