Problem 20 - 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. \(\frac{8 c^{2}}{8 c^{2}+16 c}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{c}{c+2}\), undefined at \(c = 0\) and \(c = -2\).

Step 1: Factor the Denominator

First, identify and factor out any common factors in the denominator: The denominator is: \(8c^2 + 16c\).Factors of the denominator: \(8c\) is a common factor.Factoring the denominator gives:\[8c^2 + 16c = 8c(c + 2)\]

Step 2: Simplify the Expression

Next, simplify the rational expression by canceling out any common factors that appear in both the numerator and the denominator:The expression is:\[\frac{8c^2}{8c(c+2)}\]Cancel the common factor of \(8c\):\[\frac{8c^2}{8c(c+2)} = \frac{c}{c+2}\]

Step 3: Determine the Values for Undefined Expression

The rational expression is undefined where the denominator equals zero since division by zero is not allowed.Solve for \(c\) such that the denominator \(8c(c+2)\) is zero:Set the factored denominator equal to zero: \[8c(c+2) = 0\]This gives two conditions: 1. \(8c = 0 \Rightarrow c = 0\)2. \(c + 2 = 0 \Rightarrow c = -2\)Thus, the expression is undefined when \(c = 0\) or \(c = -2\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential step in working with rational expressions. A rational expression is in its simplest form when the numerator and denominator have no common factors other than 1.
To simplify a fraction, follow these steps:

• Find the greatest common factor (GCF) of the numerator and the denominator. In our example, both the numerator and the denominator have a common factor of \(8c\).
• Divide both the numerator and the denominator by this common factor to reduce the expression. In the problem, dividing by \(8c\) simplifies \(\frac{8c^2}{8c(c+2)}\) to \(\frac{c}{c+2}\).

This process is like breaking down numbers into their simplest components, making them easier to work with in equations and calculations.

Factoring Polynomials

Factoring polynomials is a crucial step when simplifying rational expressions. When you factor, you express a polynomial as a product of simpler polynomials. Here's how you can proceed:

• First, look for common factors. This involves identifying any terms that can be factored out from every term in the polynomial. For instance, in the denominator \(8c^2 + 16c\), the term \(8c\) is a common factor.
• Once identified, factor out the common term. The expression \(8c^2 + 16c\) simplifies to \(8c(c + 2)\).

After factoring, the polynomial becomes easier to manipulate. It's also a necessary step before cancelling terms between the numerator and the denominator.

Undefined Expressions

Rational expressions can become problematic when their denominator equals zero. Division by zero is undefined in mathematics. To determine when a rational expression is undefined:

• Set the denominator equal to zero and solve for the variable. Taking \(8c(c+2) = 0\) from our example, solve each part: \(8c = 0\) gives \(c = 0\), and \(c + 2 = 0\) leads to \(c = -2\).
• The values of \(c = 0\) and \(c = -2\) are where the expression becomes undefined.

Identifying these values ensures that you handle expressions correctly, avoiding undefined operations in your calculations.