Problem 10 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$7 x-2 y$$

Step-by-Step Solution

Answer

The degree of the polynomial is 1.

Step 1: Identify the Terms

A polynomial is made up of one or more terms which consist of a coefficient and variables raised to an exponent. The given polynomial is composed of two terms: 7x and -2y.

Step 2: Determine the Degree of Each Term

The degree of a term is the sum of the exponents of the variables in the term. For the term 7x, the degree is 1 because the exponent of x is 1. For the term -2y, the degree is 1 because the exponent of y is 1.

Step 3: Find the Degree of the Polynomial

The degree of the polynomial is the highest degree of its terms. Both terms 7x and -2y have a degree of 1. Therefore, the degree of the polynomial is 1.

Key Concepts

Polynomials

Polynomials are expressions that consist of variables and coefficients, combined using operations like addition, subtraction, and multiplication. An easy way to identify them is by looking for mathematical expressions that include terms such as \(ax^n\), where \(a\) is a coefficient, \(x\) is a variable, and \(n\) is a non-negative integer representing the exponent. Polynomials can be classified based on the number of terms they have:

• Monomials: Consist of one term, such as \(3x^2\).
• Binomials: Consist of two terms, like in our example, \(7x - 2y\).
• Trinomials: Have three terms, an example would be \(x^2 + 4x + 4\).

Understanding polynomials is essential as they form the foundation of many algebraic concepts and are widely used in calculus and higher mathematics. Each polynomial can be broken down into simpler units called terms that determine its characteristics, which leads us to our next topic.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They often serve as a bridge to polynomials, as they can take various forms, including polynomial expressions. When distinguishing between them, it’s important to note:

• Constants: Numbers without variables, like \(5\), are also part of algebraic expressions.
• Variables: Symbols that represent an unknown value, such as \(x\) or \(y\). They are fundamental components.
• Operators: These are symbols like \(+\), \(-\), \(\times\), and \(\div\) used to combine numbers and variables.

An algebraic expression becomes a polynomial when the exponents of its variables are non-negative integers. For instance, our exercise expression \(7x - 2y\) fits the criteria as it consists of two variables raised to the power of 1. Understanding algebraic expressions allows students to evaluate, simplify, and interpret mathematical models in real-world scenarios.

Terms in Polynomials

A polynomial is made up of individual components known as terms. Each term is composed of a coefficient and a variable raised to a specific exponent. Examining the terms is a crucial step in analyzing polynomials. Here's how it works:

• Coefficient: This is the numerical factor in a term. In \(7x\), \(7\) is the coefficient.
• Variable and Exponent: Variables are raised to a power called an exponent. For \(7x\), the variable \(x\) has an exponent of 1.
• Degree of a Term: Calculated by adding together the exponents of all the variables in the term. It helps in determining the term's contribution to the polynomial's overall degree.

In our example \(7x - 2y\), both terms, \(7x\) and \(-2y\), have variables raised to the power of 1, making the polynomial degree 1. Understanding the structure of each term allows students to comprehend more complex polynomials and their behavior.