Problem 27 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{5 x^{2}+7}{10 x}\)

Step-by-Step Solution

Answer

The expression \(\frac{5x^2 + 7}{10x}\) is already in simplest form.

Step 1: Factor the Numerator and Denominator

To simplify the rational expression \(\frac{5x^2 + 7}{10x}\), we first look to factor both the numerator \(5x^2 + 7\) and the denominator \(10x\). However, the numerator is a sum of terms, and does not have any common factors other than 1. The denominator \(10x\) can be written as \(5 \cdot 2x\).

Step 2: Identify Common Factors

Inspect the nominator and the denominator for any common factors. Notice that neither the term 5 in the numerator nor the terms 2 and \(x\) in the denominator is shared. The expressions have no common factors that can be cancelled.

Step 3: Simplify the Expression if Possible

Since there are no common factors, the expression \(\frac{5x^2 + 7}{10x}\) cannot be simplified further. It reflects the simplest form of the given rational expression.

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex polynomial into simpler, multiplied expressions, or factors, that when multiplied together, give the original polynomial. This process is crucial for simplifying rational expressions as it helps in identifying common factors.

• Start by searching for a common factor in all terms of the polynomial. A common factor is a number, variable, or combination of both that divides each term without leaving a remainder.
• In the given problem, the numerator \(5x^2 + 7\) cannot be factored further since it does not have a common factor other than 1.
• The denominator \(10x\) can be expressed as the product \(5 \cdot 2x\), showing its factors clearly.

Factoring can be more complex with polynomials of higher degrees or more terms, requiring different methods like grouping or using the quadratic formula.

Simplest Form

The simplest form of a rational expression is achieved when there are no more common factors to cancel between the numerator and the denominator. This form is the most reduced version of the expression, making it easier to understand and work with in equations.

• After factoring both the numerator and the denominator, compare their factors to check for any commonalities.
• In the example \(\frac{5x^2 + 7}{10x}\), there are no common factors between the numerator and denominator to cancel.
• Therefore, this is already in its simplest form.

Simplifying a rational expression to its simplest form can often make complex algebraic problems more manageable and helps reveal other mathematical properties.

Algebraic Expressions

Understanding algebraic expressions is essential when working with rational expressions because it involves a mix of numbers, variables, and operations. In algebra, these expressions represent values in equations and require manipulation using specific rules.

• An algebraic expression can be as simple as a single term like \(x\) or as complex as a multi-term polynomial.
• They can include operations such as addition, subtraction, multiplication, and division.
• For the expression \(\frac{5x^2 + 7}{10x}\), the upper part, \(5x^2 + 7\), and the lower part, \(10x\), make the whole an algebraic fraction.

Being comfortable with these expressions and knowing how to manipulate them is crucial for solving mathematical equations and simplifying fractions effectively.