Problem 21 - Beginning and Intermediate Algebra

Question

In each term, give the numerical coefficient. \(3 \mathrm{~m}^{2}\)

Step-by-Step Solution

Answer

3

Step 1: Identify the term

Observe the given term, which is: \(3 \, \mathrm{m}^2\).

Step 2: Determine the numerical coefficient

Identify the numerical part of the term. The numerical coefficient is the number that multiplies the variable part. In this term, the numerical coefficient is \(3\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like + or -). They can be as simple as a single number or a combination of terms. For example, in the expression 3m² + 2m - 5, there are three terms: 3m², 2m, and -5. These terms are combined using addition and subtraction. Algebraic expressions are essential because they allow us to describe and work with relationships between numbers and variables in a compact form. Let's break it down:

• Numbers are constants without variables, e.g., 5 or -7.
• Variables represent unknown values and are usually denoted as letters like m or x.
• Operators are symbols such as +, -, *, and /, used to perform operations on numbers and variables.

Together, these components form algebraic expressions that can be simplified, evaluated, and factored depending on the problem to be solved.

Polynomials

Polynomials are a special kind of algebraic expression where each term is formed by multiplying a constant and one or more variables raised to non-negative integer powers. For instance, the expression 4x³ + 3x² - 2x + 1 is a polynomial. Let's look at its features:

• Each term in a polynomial is called a 'monomial'.
• The highest power of the variable in a polynomial is called its 'degree'. In our example, the degree is 3 because the highest power of x is x³.
• Polynomials can have constants, variables, and exponents, but no division by a variable.

Polynomials are widely used in math and science to model different situations, such as trajectories in physics or growth rates in biology.

Coefficients

A coefficient is the numerical part of a term in an algebraic expression or polynomial. It tells us how many times the term is taken. For example, in the term 3m², the coefficient is 3. Coefficients are important because they impact the values of the terms and thus the entire expression or polynomial. Here’s how to identify coefficients:

• Look at each term in the polynomial or expression.
• The coefficient is the number in front of the variable(s).

Even if only a single term is present, as in the term 3m², the coefficient is evident. It's fundamentally necessary because it phases to the scalar multiplication of variables, impacting the overall outcome profoundly when these equations are solved or simplified.