Problem 12 - Intermediate Algebra

Question

Simplify each rational expression, if possible. a. \(\frac{x+8}{x}\) b. \(\frac{3 a^{2}+23}{a^{2}}\)

Step-by-Step Solution

Answer

Both expressions are already simplified; no further simplification is possible.

Step 1: Identify and Simplify the Expression

For \[a. \frac{x+8}{x} \]there is no common factor in the numerator and the denominator other than 1. Therefore, this expression is already simplified as much as possible, and it can't be reduced any further.

Step 2: Simplify the Second Expression

Consider the expression in part b:\[b. \frac{3a^{2}+23}{a^{2}} \]First, check if there's any common factor or if the numerator can be factored. However, \(3a^{2} + 23\) does not factorize further with \(a^2\). Since there's no common factor in \(3a^{2} + 23\) and \(a^{2}\) other than 1, this expression is also in its simplest form.

Key Concepts

Factors in Algebra

Factors in algebra play a crucial role in simplifying expressions and solving equations. A factor is a number or an expression that divides another number or expression evenly—without any remainder. In algebra, factors can be both numbers and variables. Consider the expression \(x+8\). Here, each term is separate, and there are no common factors other than 1. For example, if we have \(3a^2 + 23\), neither term shares a factor with \(a^2\) besides 1. This means their greatest common factor is 1 and they cannot be simplified further. In order to factor an algebraic expression, we usually look for:\

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• Common terms that can be pulled out (greatest common factor approach).
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• Patterns such as difference of squares or perfect square trinomials.
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\Recognizing these patterns helps us in rewriting expressions in simpler forms, which can be crucial for simplification tasks.

Rational Expressions

Rational expressions are similar to fractions, but they contain polynomials in the numerator and the denominator. A rational expression looks like this: \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. Just like fractions, if the numerator and the denominator have a common factor, the expression can sometimes be simplified. When dealing with rational expressions, it is essential to understand that the denominator must not be zero, since division by zero is undefined. We often look for common factors in the numerator and denominator to simplify the expression, which is similar to reducing numerical fractions.Here are some points about radical expressions:
- Simplification involves canceling out identical terms in both the numerator and denominator.- Always check the denominator for zero restrictions. - Sometimes, expressions cannot be simplified further if the polynomials do not share any common factors. This is the case with expressions like \( \frac{x+8}{x} \) and \( \frac{3a^2 + 23}{a^2} \) given in the example.

Expression Simplification

Expression simplification involves rewriting an expression in a form that is as simple as possible. The aim is to reduce complexity without changing the value of the expression. For rational expressions, it's about making them easier to understand or work with, by taking out common factors or applying algebraic properties. Here's a step-by-step approach to simplifying expressions:

• Identify and mark all common factors in both the numerator and the denominator.
• Cancel out these common factors to simplify the expression.
• Ensure no factors remain that can be further simplified.

In the examples given, neither \( \frac{x+8}{x} \) nor \( \frac{3a^2 + 23}{a^2} \) can be simplified because all terms have no common factors except for 1.While simplifying, it's important to remember that expressions need to be equivalent before and after simplification, ensuring no loss of value throughout the process. This is also where one might check for extraneous solutions, which do not solve the original equation. Each expression simplified is checked to ensure it remains valid in all real number scenarios except where the original constraints (like denominator becoming zero) apply.