Problem 1 - Elementary Algebra

Question

For Problems \(1-8\), determine the degree of each polynomial. $$ 7 x^{2} y+6 x y $$

Step-by-Step Solution

Answer

The degree of the polynomial is 3.

Step 1: Identify Individual Terms

A polynomial consists of terms that are combinations of variables and coefficients. In this exercise, the polynomial given is \(7x^2y + 6xy\). This polynomial has two terms: \(7x^2y\) and \(6xy\).

Step 2: Determine the Degree of Each Term

The degree of a term in a polynomial is the sum of the exponents of all variables in that term. For the term \(7x^2y\), the degree is \(2 + 1 = 3\). For the term \(6xy\), the degree is \(1 + 1 = 2\).

Step 3: Identify the Degree of the Polynomial

The degree of a polynomial is the highest degree among its terms. We have determined that the term \(7x^2y\) has a degree of 3, and the term \(6xy\) has a degree of 2. Therefore, the degree of the polynomial \(7x^2y + 6xy\) is 3.

Key Concepts

Polynomial Terms

In the realm of polynomials, the term "polynomial terms" refers to individual components that come together to form a polynomial expression. Each term is a combination of coefficients (numbers) and variables raised to certain powers (exponents). For instance, in the polynomial \(7x^2y + 6xy\), each part (\(7x^2y\) and \(6xy\)) that is added together is considered a separate term.

Every term within a polynomial has three main components:

• The coefficient, which is the numerical part, like 7 or 6 in our example.
• Variables, like \(x\) and \(y\), which are the unknowns we often solve for.
• Exponents, which indicate how many times the variable is used in multiplication.

Understanding terms is essential because they serve as the basic building blocks of polynomial expressions. When we talk about manipulating polynomials, we typically refer to altering and organizing these terms.

Polynomial Expressions

A polynomial expression is a mathematical phrase that can have constants, variables, and exponents. Polynomials are made up of one or more polynomial terms. Taking the previous example \(7x^2y + 6xy\), this entire combination forms a single polynomial expression.

Here are some basics of polynomial expressions:

• They can be made of just one term, known as a monomial, or several terms, such as a binomial (two terms) or trinomial (three terms).
• The operations involved are typically addition, subtraction, or multiplication of terms.
• Polynomials can be organized by their degree, which is the highest degree of its terms.

Polynomial expressions are foundational in algebra due to their flexibility in modeling complex problems. They are essential in solving equations and understanding functions across many areas of mathematics and applied sciences.

Algebraic Degree

The algebraic degree of a polynomial is a crucial concept as it provides the highest power of the variables present in any one of the polynomial's terms. This is not just a mere number; it gives important information about the polynomial's behavior and the "strength" of the terms. In our example \(7x^2y + 6xy\), the degree of the entire polynomial expression is 3.

Here's how to determine the degree of a polynomial:

• Look at each term separately and add up the exponents for all variables in that term. This sum is that term's degree.
• From these values, the degree of the polynomial is the highest degree found among these terms.

Recognizing the algebraic degree of a polynomial helps in understanding its graphing potential and solving capabilities. Functions derived from polynomials of higher degrees, for instance, may have more roots or extrema, providing a more complex graph.