Problem 6 - Algebra A Combined Function

Question

Find the degree of each polynomial and determine whether it is a monomial, binomial, trinomial, or none of these. See Examples 2 and 3. $$ -6 y+4 $$

Step-by-Step Solution

Answer

The polynomial is a binomial with a degree of 1.

Step 1: Identify the Terms in the Polynomial

The given polynomial is \(-6y + 4\).To determine its classification, we first identify the number of terms. A term is any part of the polynomial that is separated by a plus (+) or minus (-) sign. In this polynomial, the terms are \(-6y\) and \(4\). This means the polynomial has two terms.

Step 2: Classify the Polynomial by Number of Terms

Based on the number of terms identified:- A polynomial with one term is called a monomial.- A polynomial with two terms is called a binomial.- A polynomial with three terms is called a trinomial.Since the given polynomial \(-6y + 4\) has two terms, it is classified as a binomial.

Step 3: Find the Degree of Each Term

To find the degree of the polynomial, we first find the degree of each term.- The term \(-6y\) has a degree of 1, because the variable \(y\) is raised to the power of 1.- The term \(4\) is a constant and does not have any variables, so its degree is 0.

Step 4: Determine the Degree of the Polynomial

The degree of a polynomial is determined by the term with the highest degree:- In \(-6y + 4\), the degrees are 1 (from \(-6y\)) and 0 (from \(4\)).- Therefore, the highest degree is 1.Thus, the degree of the polynomial is 1.

Key Concepts

Degree of Polynomial

When discussing polynomials, understanding the degree of a polynomial is crucial. The degree is the highest power of the variable with a non-zero coefficient. Consider our polynomial \(-6y + 4\). Each part of a polynomial separated by a '+' or '-' sign is called a term. In this case, it consists of two terms: \(-6y\) and \(4\). The term \(-6y\) has a degree of 1 because the variable \(y\) is raised to the power of 1.
For the term \(4\), it contains no variables so its degree is 0. The overall degree of the polynomial is determined by the term with the highest degree, which in this case is 1 from the term \(-6y\). Therefore, the degree of \(-6y + 4\) is 1. Remember, the degree of a polynomial indicates how "large" or complex its variable component can get. It's a critical factor when analyzing and solving polynomial equations.

Binomial

Polynomials can be categorized by the number of terms they have. A polynomial with exactly two terms is referred to as a binomial. The prefix 'bi-' suggests the number two, similar to bicycle, which has two wheels. Having this understanding helps quickly classify polynomials based on their structural layout.
The given example, \(-6y + 4\), includes two distinct terms. These terms are separated by a '+' or '-' sign, which illustrates this concept well. Since \(-6y\) and \(4\) are the two terms forming \(-6y + 4\), this polynomial is appropriately called a binomial. Recognizing binomials is especially useful in applications like the binomial theorem, which allows for expanding expressions that are raised to a power.

Terms in Polynomial

Understanding what constitutes a term in a polynomial is fundamental. Notice that each term is a single element of the expression, separated from others by '+' or '-' signs. In the polynomial \(-6y + 4\), there are two terms: \(-6y\) and \(4\). Each one of these can include numbers, variables, or a combination of both, attached by multiplication or division.

• A single number or constant, like \(4\), stands alone and has a degree of 0 because there is no variable component.
• A variable like \(-6y\) involves multiplication with a constant and has a degree based on the power of the variable, which is 1 here.

Recognizing the different parts helps in further operations such as identifying degrees or classifying the polynomial. Always remember that separating terms correctly forms the base for all polynomial-related tasks.