Problem 53 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ 3 x^{2} y^{4}+2 x-8 y=14 $$

Step-by-Step Solution

Answer

Answer: The degree of the given polynomial equation is \(6\), and it cannot be classified as a linear, quadratic, or cubic equation because its degree is higher than \(3\).

Step 1: Identify each term

Each term in the given equation is: $$ \begin{aligned} &3x^{2}y^{4} \\ &2x \\ &-8y \\ &14 \end{aligned} $$

Step 2: Determine the degree of each term

The degree of each term is calculated by adding the exponents of the variables present in that term. Let's determine the degree of each term: 1) \(3x^{2}y^{4}\) degree: \[ 2 + 4 = 6 \] 2) \(2x\) degree: \[ 1 \] 3) \(-8y\) degree: \[ 1 \] 4) \(14\) degree: \[ 0 \]

Step 3: Identify the degree of the equation

The degree of the equation is the highest degree among all terms. We found that the degrees of the terms are \(6, 1, 1,\) and \(0\). Thus, the degree of the equation is \[ 6 \]

Step 4: Classify the equation by degree

Since the degree of the equation is \(6\), it is not a linear, quadratic, or cubic equation. Linear equations have a degree of \(1\), quadratic equations have a degree of \(2\), and cubic equations have a degree of \(3\).

Key Concepts

Degree of Polynomial

Understanding the degree of a polynomial is key to classifying the type of polynomial equation you’re dealing with. The degree is defined as the highest sum of the exponents of the variables in each term of the polynomial. For instance, in the term \(3x^{2}y^{4}\), the degree is found by adding the exponents of \(x\) and \(y\), giving us \[2 + 4 = 6\].

• The degree helps determine the behavior of a polynomial function as its variable values go to infinity.
• It also gives insight into the maximum number of roots or solutions the equation can have.

The degree of this equation is \(6\), which tells us that it is more complex than linear, quadratic, or cubic equations, classified by degrees \(1\), \(2\), and \(3\) respectively. Therefore, for our equation, we simply focus on the highest degree term, \(3x^{2}y^{4}\), which dictates its classification by degree.

Classification of Equations

Classifying equations based on their degree is a simple and systematic way of understanding the nature of an equation.

• **Linear Equations** have a degree of \(1\). They're represented by the form \(ax + b = 0\).
• **Quadratic Equations** are characterized by a degree of \(2\). They follow the form \(ax^2 + bx + c = 0\).
• **Cubic Equations** have a degree of \(3\), with the structure \(ax^3 + bx^2 + cx + d = 0\).

When dealing with equations beyond cubic, we get into the realm of higher degree polynomials such as quartic (degree \(4\)), quintic (degree \(5\)), and so forth.
In our given equation \(3 x^{2} y^{4}+2 x-8 y=14\), its degree of \(6\) means it's not a simple linear, quadratic, or cubic equation; it falls into the higher degree category, typically requiring more complex methods to solve or analyze.

Terms in a Polynomial

The building blocks of a polynomial are its terms. Each term in a polynomial consists of:

• A constant coefficient (a number multiplying the variable part),
• Variable base(s), and
• Exponents on the variables.

Let's consider the terms in our example equation: \(3x^{2}y^{4}\), \(2x\), and \(-8y\).

• The term \(3x^{2}y^{4}\) has a coefficient of \(3\), variable bases \(x\) and \(y\), and exponents \(2\) and \(4\) respectively.
• The term \(2x\) has a coefficient of \(2\) and a variable base \(x\) with an exponent of \(1\).
• The term \(-8y\) has a coefficient of \(-8\) and a variable \(y\) with an exponent of \(1\).

Understanding each term is key to determining the overall characteristics of the equation. Each term contributes to the polynomial's total degree and influences the complexity and graph of the polynomial.