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Chapter 0: Problem 10

Is the expression a monomial? If it is, name the variable(s) and the coefficient, and give the degree of the monomial. If it is not a monomial, state why not. $$ -4 x^{2} $$

Short Answer

Expert verified
-4x^{2} is a monomial. The coefficient is -4, the variable is x, and the degree is 2.

Step by step solution

01

Understand the Definition of a Monomial

A monomial is a single term algebraic expression that consists of a coefficient, variable(s) and non-negative integer exponents. For example, terms like \(3x\) or \(-2y^{3}\) are monomials.
02

Analyze the Given Expression

Examine the given expression: \(-4x^{2}\). Check if it follows the definition of a monomial. Specifically, identify if there is only one term and whether the exponents are non-negative integers.
03

Identify and Name the Components

In \(-4x^{2}\), -4 is the coefficient and x is the variable. The exponent of x is 2, which is a non-negative integer.
04

Determine if the Expression is a Monomial

Since \(-4x^{2}\) consists of a single term with a non-negative integer exponent, it meets the definition of a monomial.
05

State the Coefficient, Variables, and Degree

The coefficient is -4, the variable is x, and the degree of the monomial is the sum of the exponents of the variables, which in this case is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

coefficient
In algebra, the coefficient is the numerical part of a term that contains both numbers and variables. For instance, in the term \(-4x^{2}\), -4 is the coefficient.
It's important to recognize coefficients because:
  • They tell you how many times the variable(s) are multiplied.
  • Identifying the coefficient helps simplify and manipulate expressions in algebra.
In our specific example, -4 is the coefficient, which means -4 times the variable \(x^{2}\).
variable
A variable is a symbol used to represent an unknown value in mathematical expressions and equations. Common variables include x, y, and z.
In the expression \(-4x^{2}\), x is the variable.
Variables allow us to create general formulas and equations that can be solved for different values. Identifying the variable is essential because:
  • They help us understand which parts of the expression can change.
  • It鈥檚 crucial for solving equations and simplifying expressions.
In our example, x is raised to the power of 2, indicating it's a squared variable.
degree of a monomial
The degree of a monomial is the sum of the exponents of all its variables. It tells you the highest power of the variable in the expression.
For example, in the term \(x^{2}\), the degree is 2. In the expression \(-4x^{2}\), the degree is also 2 because we only have one variable, x, raised to the power of 2.
Understanding the degree of a monomial is helpful because:
  • It indicates the complexity of the term.
  • It鈥檚 used in advanced algebra for polynomial operations.
In our example, the degree is simply the exponent on the variable, which is 2.
non-negative integer exponent
A non-negative integer exponent is an exponent that is either a positive whole number or zero. In the expression \(x^{2}\), 2 is a non-negative integer exponent.
Monomials require non-negative exponents:
  • Negative exponents would disqualify the expression as a monomial.
  • Non-integer exponents would also disqualify it.
For example, \(-4x^{2}\) is valid because the exponent 2 is a non-negative integer.
This requirement ensures that monomials have a standard form suitable for algebraic manipulation.
single term
A single term in algebraic expressions is a combination of numbers and variables without any addition or subtraction separating them. For an expression to be a monomial, it must consist of only one term.
Expressions like \(-4x^{2}\) and \(3xy\) are monomials with single terms.
Single terms are straightforward:
  • They simplify mathematical operations.
  • Manipulations involving monomials are often easier to perform compared to polynomials.
In our example, the expression \(-4x^{2}\) is a single term and thus qualifies as a monomial.

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Most popular questions from this chapter

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-3}{\sqrt{5}+4}$$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}}$$

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$3 x^{-1 / 2}+\frac{3}{2} x^{1 / 2} \quad x>0$$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{5}{\sqrt{2}-1}$$

Use a calculator to approximate each radical. Round your answer to two decimal places. $$\frac{\sqrt{5}-2}{\sqrt{2}+4}$$
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