Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{2 x^{6}+3 x^{3}}{2 x}$$

Step-by-Step Solution

Answer

The simplified expression is \( x^5 + \frac{3x^3}{2x} \).

Step 1: Factor the Numerator

Look at the numerator, which is \(2x^6 + 3x^3\). Factor out the greatest common factor, which in this case is \(x^3\). This makes the expression \(x^3(2x^3 + 3)\).

Step 2: Rewrite Expression Using Division Property

Apply the division property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) to rewrite the expression as follows: \[\frac{x^3(2x^3+3)}{2x} = x^3 \left( \frac{2x^3}{2x} + \frac{3}{2x} \right)\].

Step 3: Simplify Each Term Separately

Simplify each term within the parentheses. For \(\frac{2x^3}{2x}\), divide both the coefficient and the variable: \[ \frac{2x^3}{2x} = x^2 \]. For \(\frac{3}{2x}\), simplify the fraction as it cannot be reduced further: \[\frac{3}{2x}\].

Step 4: Combine Simplified Terms

Combine the simplified terms to get the final expression: \[ x^3 \left( x^2 + \frac{3}{2x} \right) \]. This is the simplified form of the original rational expression.

Key Concepts

Factoring

Factoring is a crucial step in simplifying rational expressions. It involves identifying and extracting common factors from terms. When you look at the expression in the numerator, such as the term \(2x^6 + 3x^3\), the goal is to simplify it by finding a factor common to both. In this case, the greatest common factor is \(x^3\). By factoring it out, the expression becomes \(x^3(2x^3 + 3)\). This makes it easier to simplify the whole rational expression later.

• Factoring reduces the complexity of terms.
• It is essential in preparing terms for division or cancellation.

Remember, factoring is the friendly "unpacking" of terms to make them easier to work with.

Division Property

The division property is highlighted here with the formula \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) which shows us how to divide a sum by a single term. We apply it to the rational expression like this: \(\frac{x^3(2x^3 + 3)}{2x} = x^3 \left( \frac{2x^3}{2x} + \frac{3}{2x} \right)\).

This property helps in separating out terms within the fraction, allowing for easier simplification.

• Remember, each term in the numerator needs to be divided by the denominator.
• Ensure that all terms are simplified separately before combining them back together.

This method makes solving expressions more straightforward by breaking them down into valuable parts.

Rational Expressions

Rational expressions, much like fractions, involve a numerator and a denominator. In these expressions, variables appear alongside or instead of numbers. The main task is to simplify these expressions as much as possible.
In our example, \(\frac{2x^6+3x^3}{2x}\), is broken down by factoring and applying properties like division.

• Simplifying these expressions often requires multiple steps like factoring and division.
• The goal is to reduce the expression to its simplest form without altering its value.

With rational expressions, practice and patience are key, as well as familiarity with algebraic principles. Keep in mind these expressions can describe many relationships in math and science.

Greatest Common Factor

The Greatest Common Factor (GCF) is used to simplify algebraic expressions and is an integral part of factoring. It's the largest factor shared by all terms in the expression. When dealing with \(2x^6 + 3x^3\), identifying the GCF helps in breaking down the terms. Here, it is \(x^3\), meaning both terms share at least three powers of \(x\).

• Finding the GCF effectively reduces the expression's complexity.
• Always look for the GCF first when simplifying rational expressions.

This approach reduces large polynomial terms into manageable parts, setting the stage for easier manipulation and understanding of expressions.