Problem 1 - Algebra Form and Function

Question

Which of the expressions in Exercises \(1-6\) are equivalent to monomials in \(x ?\) $$ -\frac{x^{3}}{5} $$

Step-by-Step Solution

Answer

Answer: Yes, the expression \(-\frac{x^3}{5}\) is equivalent to a monomial in \(x\).

Step 1: Identify the expression

We are given the following expression: $$ - \frac{x^3}{5} $$ Our task is to determine if it is equivalent to a monomial in \(x\).

Step 2: Check for a single term

To be a monomial, the expression must have only one term. We can observe that our given expression has only one term: $$ - \frac{x^3}{5} $$

Step 3: Check the exponent

Now, we need to check whether the expression has a non-negative integer exponent. From the given expression, the exponent is \(3\), which is a non-negative integer: $$ x^3 $$

Step 4: Conclude if the expression is a monomial

Since the given expression, \(- \frac{x^3}{5}\), has only one term and its exponent is a non-negative integer, we can conclude that it is equivalent to a monomial in \(x\).

Key Concepts

Single Term Expressions

Monomials are considered single term expressions. They consist of only one term, which can be a number, a variable, or a product of numbers and variables raised to a power. In mathematics, having a single term simplifies calculations and analysis, making it crucial for simplifying algebraic expressions.
Understanding what constitutes a single term is essential in determining if an expression is a monomial. In our given problem,

• The expression \(-\frac{x^3}{5}\) contains only one term: \(-\frac{x^3}{5}\).
• There's no addition or subtraction separating different parts of the expression.
• The division sign does not create a new term; it modifies the term as a whole.

This demonstrates how this expression fits the criteria of a monomial as a single term expression.

Exponents

Exponents play a vital role in mathematics, especially when dealing with monomials. They represent repeated multiplication of a base number or variable. In the context of a monomial, an exponent tells us how many times to multiply the variable by itself.
For our expression \(-\frac{x^3}{5}\), the exponent of the variable \(x\) is 3:

• This means \(x\) is multiplied by itself three times, represented as \(x \times x \times x\).
• Exponents must be non-negative integers when determining if an expression qualifies as a monomial.

Moreover, higher exponents increase the degree of a monomial, which can have implications in algebraic operations and calculus.

Non-negative Integers

A key feature of monomials is that the exponents of their variables must be non-negative integers. That means they can be any whole number that is zero or higher, including numbers like 0, 1, 2, 3, and so on. This is because negative exponents or fractional exponents would change the nature of the expression from a polynomial to something else.
In our problem, the exponent \(3\) meets these criteria:

• The number 3 is a whole number and greater than 0.
• It allows the expression \(-\frac{x^3}{5}\) to remain in the category of monomials.

The restriction to non-negative integers ensures the expression stays well within the realm of polynomial functions, which are crucial in various fields such as calculus, physics, and engineering.