Problem 49 - Algebra A Combined Function

Question

Identify the degrees of the terms and the degree of the polynomial. See Example 12. $$ x^{3} y-6+2 x^{2} y^{2}+5 y^{3} $$

Step-by-Step Solution

Answer

The degree of the polynomial is 4.

Step 1: Identify each term

The given polynomial is \( x^{3}y - 6 + 2x^{2}y^{2} + 5y^{3} \). It consists of the terms \( x^{3}y \), \(-6\), \(2x^{2}y^{2}\), and \(5y^{3}\).

Step 2: Determine the degree of each term

For \( x^{3}y \), the degree is \(3 + 1 = 4\). For the constant term \(-6\), the degree is 0 since it has no variable part. For \( 2x^{2}y^{2} \), the degree is \(2 + 2 = 4\). Finally, for \( 5y^{3} \), the degree is 3.

Step 3: Determine the degree of the polynomial

The degree of the polynomial is the highest degree among all its terms. From Step 2, we see that the highest degree is 4, which occurs in the terms \( x^{3}y \) and \( 2x^{2}y^{2} \).

Key Concepts

Terms in Polynomial

In mathematics, a polynomial is an expression consisting of variables, coefficients, and arithmetic operations such as addition, subtraction, and multiplication. Each individual component of a polynomial, separated by plus (+) or minus (-) signs, is known as a term.

In the provided polynomial \(x^{3}y - 6 + 2x^{2}y^{2} + 5y^{3}\), the terms are:

• \(x^{3}y\)
• \(-6\)
• \(2x^{2}y^{2}\)
• \(5y^{3}\)

Each term is composed of a coefficient and possibly a variable part. Understanding these terms individually helps in analyzing the polynomial effectively.

Degree of a Term

The degree of a term in a polynomial is crucial to understanding the polynomial's characteristics. It represents the sum of the exponents of the variables in that term. This is helpful in determining how the expression behaves for larger values of the variables.

For instance, in:

• \(x^{3}y\), the degree is calculated as \(3 + 1 = 4\)
• \(-6\), which, being a constant, has no variables, thus a degree of 0
• \(2x^{2}y^{2}\), the degree is \(2 + 2 = 4\)
• \(5y^{3}\), the degree is simply 3

Observing these degrees helps us quickly understand which terms in the polynomial are more dominant when it comes to the function's behavior.

Constant Term

In a polynomial, a constant term is a term which does not include any variables. This means it stays unchanged no matter the values of other variables in the polynomial. Constant terms are particularly simple since their degree is always zero.

Looking at our polynomial, the constant term is \(-6\). Despite its simplicity, this term may significantly affect the polynomial's value, especially for smaller inputs of variables where the effect of other terms diminishes.

Highest Degree of Polynomial

The highest degree of a polynomial is defined as the largest degree among all its terms. It describes the overall behavior and category of the polynomial, thus, knowing it is essential for understanding how the polynomial graphically behaves as variable values increase or decrease.

For the given polynomial \(x^{3}y - 6 + 2x^{2}y^{2} + 5y^{3}\), we identified the terms \(x^{3}y\) and \(2x^{2}y^{2}\), both having the highest degree of 4. This tells us that the polynomial itself has a degree of 4, dictating its status as a quartic polynomial, which typically forms a curve with certain symmetrical properties.