Problem 4 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$5 x^{3} y^{2}-6 x^{3} y^{3}$$

Step-by-Step Solution

Answer

The degree of the polynomial is 6.

Step 1: Define the Polynomial Terms

The given polynomial is composed of two terms: \(5 x^{3} y^{2}\) and \(-6 x^{3} y^{3}\). Identify each term separately.

Step 2: Find the Degree of Each Term

To find the degree of each term, add the exponents of the variables in each term.\- For \(5 x^{3} y^{2}\), the degree is \(3 + 2 = 5\).\- For \(-6 x^{3} y^{3}\), the degree is \(3 + 3 = 6\).

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is determined by the term with the highest degree. Compare the degrees found in Step 2. Since the term \(-6 x^{3} y^{3}\) has the highest degree of 6, the polynomial's degree is 6.

Key Concepts

Polynomial Terms

Understanding polynomial terms is the first step in analyzing polynomials. In any polynomial expression, terms are the individual components that make up the polynomial. For example, in the polynomial given, which is \(5 x^{3} y^{2} - 6 x^{3} y^{3}\), there are two separate terms. Each term is made up of coefficients and variables, where:

• The coefficient is the numerical part, like 5 and -6.
• The variables are usually represented by letters like \(x\) and \(y\).

Each term is essentially a product of a numerical coefficient and variables raised to certain powers. The concepts of polynomial terms lay the groundwork for exploring further properties, such as degrees.

Exponents

Exponents play a critical role in determining the characteristics of polynomial terms. An exponent in a polynomial term indicates the power to which the variable is raised. For example, in the term \(5 x^{3} y^{2}\), the exponents are 3 and 2 for \(x\) and \(y\) respectively. What this means is:

• \(x^{3}\) implies \(x\) is multiplied by itself three times.
• \(y^{2}\) means \(y\) is multiplied by itself twice.

Exponents not only define the order of the variables, but also help in calculating the degree of a term, which is crucial for understanding the degree of an entire polynomial. In mathematical operations involving polynomials, paying attention to the exponents is key to accurate solutions.

Highest Degree Term

The highest degree term in a polynomial determines the overall degree of the polynomial. The degree of a term is calculated by adding up the exponents of all variables within that term. For example, with the polynomial \(5 x^{3} y^{2} - 6 x^{3} y^{3}\):

• The first term \(5 x^{3} y^{2}\) has a degree of \(3 + 2 = 5\).
• The second term \(-6 x^{3} y^{3}\) has a degree of \(3 + 3 = 6\).

Among these, the second term has the highest degree, which is 6. Therefore, the degree of the entire polynomial is determined as 6. It is important to remember that in cases with multiple terms, the highest degree term dictates the polynomial's degree, providing insight into the behavior and properties of the polynomial as a whole.