Problem 11 - Algebra 2 and Trigonometry

Question

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{6}{10}\)

Step-by-Step Solution

Answer

The simplest form is \( \frac{3}{5} \) and the expression is defined for all variables.

Step 1: Find the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{6}{10} \), we need to find the greatest common divisor (GCD) of the numerator (6) and the denominator (10). Here, the GCD of 6 and 10 is 2.

Step 2: Divide Numerator and Denominator by the GCD

Divide both the numerator and the denominator of the fraction \( \frac{6}{10} \) by the GCD, which is 2: \( \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \).

Step 3: Identify Values That Make the Fraction Undefined

A fraction is undefined if the denominator is zero. Here, the denominator 10 is not zero for any value, so there are no undefined values in this context.

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is a useful tool when simplifying fractions. It is the largest positive integer that divides two numbers without leaving a remainder. In our given exercise, we needed to simplify the fraction \( \frac{6}{10} \). To do this, we identified the GCD of the numerator (6) and the denominator (10).

Here's how you can determine the GCD:

• List the factors of each number. For 6, the factors are 1, 2, 3, and 6. For 10, the factors are 1, 2, 5, and 10.
• Identify the common factors. In this case, both 6 and 10 share the factors 1 and 2.
• The greatest of these common factors is the GCD. So, for 6 and 10, the GCD is 2.

Using the GCD helps in simplifying fractions more conveniently by ensuring that the fraction is reduced to its simplest form. By dividing both the numerator and the denominator by their GCD, we ensure the resulting fraction, \( \frac{3}{5} \), is in the simplest form.

Undefined Fractions

An undefined fraction occurs when the denominator is zero. This is because division by zero is undefined in mathematics. The main point here is to check the denominator when simplifying any rational expression to determine if there are any values that could make the fraction undefined.

In our exercise with the fraction \( \frac{6}{10} \), the original denominator is 10. Since 10 is never equal to zero, the fraction remains defined for all real numbers. But imagine the denominator was expressed in terms of a variable, such as \( x + 10 \). We would look for values of \( x \) that make this expression equal zero:

• Set the expression in the denominator to zero: \( x + 10 = 0 \).
• Solve for \( x \) to find values where the expression becomes undefined: \( x = -10 \).

This approach is critical in algebra for identifying potential restrictions on variable values that make fractions undefined.

Numerator and Denominator

In rational expressions, the numerator and the denominator are essential parts that define the expression. The numerator is the top part of the fraction, while the denominator is the bottom part. Understanding their roles helps in simplifying and analyzing rational expressions.

For our exercise, the fraction \( \frac{6}{10} \) has a numerator of 6 and a denominator of 10. The goal of simplifying is to find a common factor that can be divided from both parts to reduce the fraction's complexity. By dividing each by the GCD of 2, we achieved \( \frac{3}{5} \), where 3 is the new numerator and 5 is the new denominator.

Key points to remember:

• If the numerator is zero and the denominator is non-zero, the fraction equals zero.
• The roles of numerator and denominator are crucial in determining the fraction’s value and potential undefined values.

Examining both parts carefully ensures accurate simplification and understanding of rational expressions.