Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{b+1}{4} \cdot \frac{12}{5 b+5} $$

Step-by-Step Solution

Answer

The product, in simplest form, is \( \frac{3}{5} \), and it is undefined for \( b = -1 \).

Step 1: Factor the Denominators

Identify and factor any expressions in the denominators. Here, the expression \(5b + 5\) in the denominator of the second fraction can be factored as \(5(b + 1)\).

Step 2: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together:\[\frac{b+1}{4} \cdot \frac{12}{5(b+1)} = \frac{(b+1) \cdot 12}{4 \cdot 5(b+1)}.\]

Step 3: Simplify the Expression

Before multiplying the fractions, notice that \(b+1\) appears in both the numerator and the denominator. They can be cancelled out, resulting in:\[\frac{12}{4 \cdot 5} = \frac{12}{20}.\]Simplify \(\frac{12}{20}\) by finding the greatest common divisor (4):\[\frac{12 \div 4}{20 \div 4} = \frac{3}{5}.\]

Step 4: Identify Restrictions for Variables

The original expression had \(b+1\) in the denominators of one of the fractions. Since division by zero is undefined, find when \(b+1=0\):\[b = -1\].Thus, the expression is not defined for \(b = -1\).

Key Concepts

Factoring

Factoring is a crucial step in managing algebraic expressions, especially when dealing with fractions. In algebra, factoring involves rewriting an expression as a product of its factors. This can be particularly helpful when trying to simplify expressions. In our exercise, the expression in the denominator \(5b + 5\) is simplified by recognizing it as a common factor: \(5(b + 1)\).
This approach helps to see where terms can cancel out when simplifying fractions. Recognizing a common factor is the key to unlocking a simpler form of the expression.
This method requires practice and an eye for noticing common terms or numerical factors within expressions. Establishing this habit can greatly simplify solving algebraic fraction problems.

Multiplying Fractions

Multiplying fractions requires multiplying the numerators and denominators separately. This step might seem straightforward but sets the foundation for the rest of the simplification. In algebra, maintaining structure while multiplying ensures precision and clarity.
In our example, we combine the numerators \((b+1)\cdot 12\) and the denominators \(4\cdot 5(b+1)\). Multiplying fractions this way keeps everything organized, making it easier to spot factors that can be cancelled out later.
It's important to consistently apply this method to keep expressions manageable and correctly structured, paving the way to simplification.

Simplifying Expressions

Simplifying expressions focuses on reducing fractions to their simplest form. The most effective way is by cancelling common factors in the numerator and denominator. This simplifies the arithmetic and algebraic complexity.
In our solution, the term \(b+1\) appears in both the numerator and the denominator, allowing us to cancel them out immediately. This process is akin to crossing out duplicate elements in both the top and bottom of the fraction, leading to a simpler expression, \(\frac{12}{20}\).
To further simplify, we find the greatest common divisor (GCD) of the remaining numbers. In this case, dividing both 12 and 20 by their GCD, 4, gives us \(\frac{3}{5}\). Practicing these steps leads to efficiency when managing fractions.

Undefined Fractions

Understanding when a fraction is undefined is crucial in algebra. An undefined fraction results from division by zero, a mathematical impossibility that disrupts calculations. Therefore, identifying potential undefined conditions is essential.
In this exercise, determining when the expression in the denominator equals zero, specifically for terms like \(b+1\), tells us which values of \(b\) make the fraction undefined. Setting \(b+1=0\) and solving gives \(b=-1\). Hence, the fraction is undefined for \(b=-1\).
This procedure ensures that you are aware of restrictions on your variable choices, highlighting the importance of evaluating denominators for zeroes throughout any algebraic fraction work.