Problem 54 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ 10 a^{2} b^{3} c^{6} d^{0} e^{4}+27 a^{3} b^{2} b^{4} b^{3} b^{2} c^{5}=1, d \neq 0 $$

Step-by-Step Solution

Answer

Based on the solution provided, classify the given polynomial equation in terms of its degree. Answer: The given polynomial equation is a 15th-degree polynomial equation.

Step 1: Identify the terms in the equation

The equation has two terms: 1. \(10a^{2}b^{3}c^{6}d^{0}e^{4}\) 2. \(27a^{3}b^{2}b^{4}b^{3}b^{2}c^{5}\)

Step 2: Compute the degree of each term

To calculate the degree of a term, we sum the exponents of all the variables in the term. 1. For the first term, the degree is \(2+3+6+0+4=15.\) 2. For the second term, we simplify it first by combining the exponents of the same variables: \(27a^{3}b^{2}b^{4}b^{3}b^{2}c^{5}=27a^{3}b^{7}c^{5}.\) The degree is \(3+7+5=15.\)

Step 3: Determine the highest degree among the terms

Both terms have a degree of 15. Therefore, the highest degree of the equation is 15.

Step 4: Classify the equation by degree

Since the highest degree of the equation is 15, which is not 1, 2, or 3, we cannot use the terms linear, quadratic, or cubic to describe it. The equation is a 15th-degree polynomial equation.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are a fundamental aspect of algebra that you will come across in many areas of math and science. An equation can be thought of as a mathematical statement that asserts the equality of two expressions. They consist of variables and coefficients, and can take on various forms, from simple linear equations to more complex polynomial equations.

When solving or classifying algebraic equations, our main goal is to understand the relationship between the variables and figure out their values. For example, in an equation such as \(10a^{2}b^{3}c^{6}d^{0}e^{4}\), we have multiple variables each raised to a power, known as an exponent. In algebra, we not only solve for these variables but also look to describe the equation based on characteristics such as its degree.

Degree of a Polynomial

The degree of a polynomial is a critical concept in understanding its characteristics and behavior. Simply put, the degree is the highest power of the variable in the polynomial. This measurement helps us determine how the polynomial will behave, especially in terms of its graph and the number of roots it can have.

To find the degree, you need to look at each term of the polynomial individually and then find the term with the largest sum of exponents. For instance, considering the term \(27a^{3}b^{2}b^{4}b^{3}b^{2}c^{5}\), we first combine the like terms, which simplifies to \(27a^{3}b^{13}c^{5}\), and add the exponents together to get \(3+13+5=21\). However, if there's a term with a higher total degree in the polynomial, that term dictates the degree of the entire polynomial.

Polynomial Terminology

When dealing with polynomials, the terminology can often be confusing. Polynomials are classified not just by their degree but also by the number of terms they contain. A single term, like \(10a^{2}b^{3}c^{6}d^{0}e^{4}\), is referred to as a monomial. If a polynomial has two terms, it is called a binomial; with three terms, it becomes a trinomial.

Furthermore, specific polynomials with degrees one, two, and three have their own names: linear, quadratic, and cubic, respectively. Higher degrees don't have special names and are simply referred to by their degree number, such as 'quartic' for fourth degree, 'quintic' for fifth degree, and so on. In our example, since the highest degree is 15, both terms must add up to this, and we label the polynomial as a 15th-degree polynomial.