Problem 20 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{7}{x^{2}}-5 x\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{7}{x^{2}} - 5x \).

Step 1: Identify the Terms

The expression is composed of two terms: \( \frac{7}{x^{2}} \) and \(-5x \). Notice one term is a fraction and the other is a monomial.

Step 2: Find a Common Denominator

To perform operations involving fractions and monomials, determine a common denominator. Here, no operation is needed between the terms since they aren't being combined directly through addition or subtraction. They are separate terms.

Step 3: Simplify Each Term

Since \( \frac{7}{x^{2}} \) cannot be simplified further without specifying values for \( x \), and \(-5x\) is already simplified, we leave them as they are. No further simplification is needed for the expression as a whole.

Step 4: Present the Final Expression

The expression, \( \frac{7}{x^{2}} - 5x \), is already in its simplest form as none of the terms require further simplification or have a common base to combine.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is like cleaning up your toolshed—everything in its proper place so that it's easy to use later on. When given an algebraic expression, look for opportunities to reduce it to its simplest form. Simplification typically involves:

• Combining like terms
• Reducing fractions if possible
• Removing or factoring out common factors

However, in the exercise above, the expression \( \frac{7}{x^{2}} - 5x \) cannot be simplified further due to the differing structures of the terms—they aren't "like" terms. This means each term must be treated individually.
Make sure to identify whether terms can be combined or simplified by examining their powers and coefficients. In our case, since they don't share a base or structure, the final expression remains as is.

Rational Expressions

A rational expression is just a fancy name for a fraction where the numerator and/or the denominator are algebraic expressions. Much like regular fractions, you can perform operations like addition, subtraction, multiplication, and division.When simplifying rational expressions, the goal is often to:

• Identify any common factors in the numerator and the denominator.
• Factor those terms out to cancel them.
• Simplify the expression afterward.

In the given exercise, \( \frac{7}{x^{2}} \) represents a rational expression. The numerator is \(7\), and the denominator is \(x^{2}\). Since 7 and \(x^{2}\) have no common factors, the expression remains simplified as is.
Keep an eye out for potential simplifications when analyzing rational expressions, but remember, not all will reduce further without additional information or specific values.

Monomials

A monomial is a single term of an expression. It can be a number, a variable, or a product of numbers and variables. They typically appear much simpler than rational expressions and don't require a denominator.In our example, \(-5x\) is a monomial, which includes:

• The coefficient: -5
• The variable: \(x\)

Monomials can be easily combined with other like monomials by adding or subtracting their coefficients. However, with the presence of a term like \( \frac{7}{x^{2}} \), the expression includes more than just monomials, meaning they cannot be simply combined as they are not like terms.
When dealing with monomials, always check if they can be simplified, combined, or factored further with others, but be aware of any surrounding terms that may impact this process.