Problem 1 - Algebra 2

Question

Determine whether each expression is a polynomial. If is a polynomial, state the degree of the polynomial. $$ 2 a+5 b $$

Step-by-Step Solution

Answer

Yes, it is a polynomial with degree 1.

Step 1: Understanding Polynomials

An expression is considered a polynomial if it consists of terms made up of constants, variables, and non-negative integer exponents of the variables. Examples of polynomial terms include \( 3x^2 \), \( -7 \), and \( 5y \).

Step 2: Analyze Each Term of the Expression

The given expression is \( 2a + 5b \). This expression has two terms: \( 2a \) and \( 5b \).

Step 3: Check Each Term for Polynomial Criteria

Both \( 2a \) and \( 5b \) are polynomial terms because they consist of a constant multiplied by a variable raised to the power of 1. The powers in both terms are integers (1 in these cases) and non-negative.

Step 4: Confirm Expression as a Polynomial

Since both terms, \( 2a \) and \( 5b \), meet the polynomial criteria (constant, variables, non-negative integer exponents), \( 2a + 5b \) is a polynomial.

Step 5: Determine the Degree of the Polynomial

The degree of a polynomial is defined as the highest power of the variable in the polynomial. In \( 2a + 5b \), each term is to the first power. Thus, the degree of the polynomial is 1.

Key Concepts

Degree of Polynomial

The degree of a polynomial is a crucial concept that helps us describe the polynomial's complexity. Think of it as the highest "power" to which any variable in the polynomial is raised. To determine the degree of a polynomial:

• Look at each term within the polynomial and find the exponents on the variables.
• The highest exponent value becomes the degree of the polynomial.

In our example, the expression is \(2a + 5b\). Each term here—\(2a\) and \(5b\)—has variables raised to the power of 1. Thus, the degree of this polynomial is 1 because both terms share the same highest exponent, which is 1. This indicates that any change in the variable has a linear effect on the value of the expression.

Polynomial Terms

In the world of polynomials, understanding the role of terms is essential. Polynomials are composed of individual parts known as "terms." A term in a polynomial is made up of:

• A constant, which is a number.
• Variables, which are letters representing numbers.
• Exponents, which tell us how many times to multiply the variable by itself.

For example, \(2a\) is a term where "2" is the constant, "a" is the variable, and the exponent is "1," since \(a^1 = a\). Similarly, in the term \(5b\), "5" is the constant, "b" is the variable, and again, it has an exponent of "1." Together, these terms form the polynomial expression \(2a + 5b\). Remember, the terms can be added or subtracted to form more complex polynomials.

Non-negative Integer Exponents

One important rule governing polynomials is that the exponents of the variables must be non-negative integers. Let's break this down:

• Non-negative integers are whole numbers that are 0 or positive, like 0, 1, 2, 3, etc.
• The exponent indicates how many times the variable is multiplied by itself. For example, in \(x^2\), the exponent 2 means \(x\) is multiplied by itself: \(x \times x\).

In the expression \(2a + 5b\), both terms \(2a\) and \(5b\) meet the criteria because the variable "a" and "b" are each raised to the power of 1. This makes them polynomial terms. If any exponent were negative or a fraction, the expression would not be considered a polynomial. Hence, ensuring exponents are non-negative integers is essential in classifying an expression as a polynomial.