Problem 1 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$7 x y+6 y$$

Step-by-Step Solution

Answer

The degree of the polynomial is 2.

Step 1: Identify Each Term of the Polynomial

The given polynomial is \(7xy + 6y\). First, we identify the individual terms of the polynomial: \(7xy\) and \(6y\).

Step 2: Determine the Degree of Each Term

The degree of a term is the sum of the exponents of the variables in that term. For \(7xy\), the exponents are 1 for \(x\) and 1 for \(y\), so the degree is \(1 + 1 = 2\). For \(6y\), the exponent for \(y\) is 1, so the degree is \(1\).

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is the highest degree among its terms. The degrees of the terms are 2 and 1. Hence, the degree of the polynomial \(7xy + 6y\) is 2.

Key Concepts

Polynomial Terms

When looking at polynomials, it's important to understand what polynomial terms are. A polynomial is made up of one or more terms. Each term is a combination of constants and variables, connected via multiplication.

Each term can include:

• Coefficients: These are the constant numbers in front of variables. For instance, in the term \(7xy\), 7 is the coefficient.
• Variables: These are the letters that represent numbers, like \(x\) and \(y\) in \(7xy\) and \(6y\).
• Exponents: Variables can have exponents, which are small numbers written above and to the right of the variable, indicating how many times the variable is multiplied by itself.

Understanding terms is a key part of comprehending polynomials since they determine factors like the polynomial's degree, a major focus of this topic.

Exponents

Exponents are a fundamental aspect of algebra, particularly when dealing with polynomials. An exponent is a number that tells you how many times to use the base number in a multiplication. In polynomials, they typically accompany variables.

For example, in the term \(7xy\), both \(x\) and \(y\) are raised to the power of 1 implicitly, even though it's not written. This means:

• \(x^1 = x\), meaning \(x\) is multiplied by itself once.
• \(y^1 = y\), meaning \(y\) is also multiplied by itself once.

For the term \(6y\), the \(y\) has an exponent of 1.

Adding exponents within a term helps determine the term's degree, which leads us to understand the overall degree of the polynomial.

Polynomial Degree Determination

The degree of a polynomial is a crucial concept when analyzing polynomials as it provides significant information about the polynomial's properties. The degree of a polynomial is determined by looking at the degrees of its individual terms.

Here's how you can determine the degree:

• First, find the degree of each term by adding up the exponents of its variables.
• For example, in the term \(7xy\), both \(x\) and \(y\) have an exponent of 1. Therefore, the degree of \(7xy\) is \(1 + 1 = 2\).
• In the term \(6y\), the degree is simply the exponent of \(y\), which is 1.

After calculating, identify the highest degree from all terms, as this will be the degree of the entire polynomial. For instance, between the terms \(7xy\) and \(6y\), the highest degree is 2, thus the degree of the polynomial \(7xy + 6y\) is 2.