Problem 58 - Algebra for College Students

Question

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{n r-6-3 n+2 r}{n r+10+2 r+5 n}\)

Step-by-Step Solution

Answer

\(\frac{r - 3}{r + 5}\) is the simplified expression.

Step 1: Rearrange and Group Terms in the Numerator

Start with the expression in the numerator: \(nr - 6 - 3n + 2r\). Rearrange the terms to make it easier to group them: \(nr - 3n + 2r - 6\).

Step 2: Factor by Grouping in the Numerator

Group the terms in the numerator: \((nr - 3n) + (2r - 6)\). Factor each group separately: \(n(r - 3) + 2(r - 3)\). Notice that \(r - 3\) is a common factor, so factor it out: \((n + 2)(r - 3)\).

Step 3: Rearrange and Group Terms in the Denominator

Now, consider the denominator: \(nr + 10 + 2r + 5n\). Rearrange the terms: \(nr + 5n + 2r + 10\).

Step 4: Factor by Grouping in the Denominator

Group the terms in the denominator: \((nr + 5n) + (2r + 10)\). Factor each group: \(n(r + 5) + 2(r + 5)\). Factor out the common \(r + 5\): \((n + 2)(r + 5)\).

Step 5: Simplify the Rational Expression

The rational expression is now \(\frac{(n + 2)(r - 3)}{(n + 2)(r + 5)}\). Cancel the common factor \(n + 2\): \(\frac{r - 3}{r + 5}\). This is the simplified form.

Key Concepts

Factoring by Grouping

Factoring by grouping can be a powerful tool when simplifying expressions, especially when dealing with polynomials. The process involves rearranging the terms of an expression and then creating groups that can be factored separately. In our example, we begin with the numerator of a rational expression. We adjust the terms to become suitable for grouping, and this requires logical ordering of expressions, a first and crucial step.

Once the terms are arranged correctly, the expression is divided into parts that have a common factor, like so:

• Group the terms involving common variables or constants together.
• Factor each group separately, pulling out common factors.
• Once grouped, identify a common binomial or monomial across the grouped terms.

After factoring the groups, look for a common factor to pull out, effectively reducing the complexity of the expression. It's similar to simplifying multiplications by finding common multiplicands.

Numerator and Denominator

In any rational expression, the two main components you'll work with are the numerator and the denominator. Understanding how to manipulate these parts is key to simplifying rational expressions.

Let's take the example \(\frac{nr - 6 - 3n + 2r}{nr + 10 + 2r + 5n}\). The numerator is the top part \(nr - 6 - 3n + 2r\), and the denominator is the lower part \(nr + 10 + 2r + 5n\). Each can be factored separately using similar techniques. When rearranging for easier grouping, patience and strategy are important:

• Rearrange terms to bring similar ones closer together.
• Balance the expression by maintaining the same order of operations.
• Factor both numerator and denominator in steps to identify common groups.

The interaction between these two components often holds the key to successfully reducing the expression. Treat each with care and look for mirroring factors on both levels.

Common Factor

Finding common factors is where the magic happens in simplifying expressions. A common factor is any quantity, be it a number or variable, that evenly divides all terms in either a part of the expression or the whole expression.

For example, consider the expression we are simplifying: \(\frac{(n + 2)(r - 3)}{(n + 2)(r + 5)}\). The common factor here is \((n + 2)\).

To identify common factors:

• Look for terms repeating across groups.
• Pull out variables or constants shared among grouped terms.
• Factor out this commonality to simplify the expression further.

Once removed, the complexity of the expression is reduced, allowing for a simpler form and clarity in understanding. The ultimate goal is to have an expression that is more manageable and reduced to its simplest form.

Rational Expressions

Rational expressions are fractions in which the numerator and the denominator are polynomials. These are akin to rational numbers but involve variables. Simplifying them involves similar rules as simplifying numerical fractions, mainly focusing on reducing common factors.

In the given example, \(\frac{nr - 6 - 3n + 2r}{nr + 10 + 2r + 5n}\), the task is to transform it into a simpler form. This involves several steps:

• Factoring the numerator and denominator.
• Finding common factors across these factored terms.
• Cancelling common factors as long as the conditions allow.

Simplifying rational expressions not only makes them easier to work with but also provides insights into the relationship between the variables involved. Always ensure that the final form is the most reduced and cannot be simplified further. This way, it's not only appropriate for further mathematical operations but also more digestible.