Problem 87 - Elementary Algebra

Question

Classify the polynomial \(4 x y+2 y\) as a monomial, binomial, or trinomial. State its degree and write the numerical coefficient of each term.

Step-by-Step Solution

Answer

Question: Classify the polynomial \(4xy + 2y\) and find its degree and the numerical coefficients of each term. Answer: The polynomial is a binomial with a degree of 2. The numerical coefficients of the terms are 4 and 2.

Step 1: Understand the terminology

A monomial is a polynomial with only one term. A binomial is a polynomial with two terms, and a trinomial is a polynomial with three terms. Terms in a polynomial are separated by addition or subtraction. The degree of the polynomial is the highest value of the sum of the exponents of its variables, and the numerical coefficient is the number multiplied by a variable or variables in a term.

Step 2: Identify the number of terms and classify the polynomial

The given polynomial is \(4xy + 2y\). There are two terms in this polynomial: \(4xy\) and \(2y\). Since there are two terms, the polynomial is classified as a binomial.

Step 3: Find the degree of the polynomial

The degree of a term is given by the sum of exponents of its variables. In the first term, \(4xy\), the exponents of \(x\) and \(y\) are both 1, resulting in a degree of 1+1=2. In the second term, \(2y\), there is only one variable \(y\) with an exponent of 1. Thus, the degree of this term is 1. Since the degree of the first term (2) is the highest, the degree of the polynomial is 2.

Step 4: Write the numerical coefficient of each term

The numerical coefficient is the number multiplied by the variables in each term. For the first term, \(4xy\), the numerical coefficient is 4. For the second term, \(2y\), the numerical coefficient is 2. So, the numerical coefficients are 4 and 2 for the two terms respectively. To summarize the results: - The polynomial is a binomial. - The degree of the polynomial is 2. - The numerical coefficients of the terms are 4 and 2.

Key Concepts

Understanding Binomials

When it comes to classifying polynomials, understanding the concept of a binomial is essential. A binomial is a type of polynomial that consists of exactly two terms. These terms are typically separated by either addition "+" or subtraction "-".
In our example, the polynomial is given as \(4xy + 2y\). You can clearly spot the two separate terms here: \(4xy\) and \(2y\). This classification is quite straightforward once you familiarize yourself with the structure of polynomials.
Polynomials can be compared to phrases in language. Understanding what a binomial is, is like recognizing a phrase that has two parts or terms combined together in mathematics. Identifying these structural elements in polynomials helps in categorizing and solving them effectively.

Deciphering Polynomial Degree

The degree of a polynomial is a bit like its power level. It's determined by looking at the term with the highest combination of exponents. In more detail, you add up all the exponents in each term, and the largest sum across all terms gives you the degree of the polynomial.
In our case, analyzing the polynomial \(4xy + 2y\), the first term \(4xy\) gives us a degree calculation. The variable \(x\) has an exponent of 1 and \(y\) too has an exponent of 1. Add these together, and you get a degree of 2.

• For the term \(4xy\), degree: \(1+1=2.\)
• For the term \(2y\), degree: \(1.\)

The term \(4xy\) with a sum of 2 contributes the highest degree. Hence, the polynomial also inherits this degree. It is crucial to identify the correct degree, as this impacts the behavior of the polynomial in equations and graphs.

Exploring Numerical Coefficients

Numerical coefficients are the constants that "lead" the variables around in each term, playing the role of multipliers. In a sense, they're the "numbers" you see directly in front of any variable in a term. In the polynomial \(4xy + 2y\), these coefficients are 4 and 2.

• In \(4xy\), the 4 is stretched along with both \(x\) and \(y\).
• In \(2y\), the 2 takes the lead role before \(y\).

Grasping the concept of these multipliers is key in simplifying and manipulating polynomials. It's like knowing how many times you count a measure in a song, as each coefficient tells you how many "copies" of its respective variable are being represented in that term. Knowing the numerical coefficient is not just a matter of pulling out numbers but plays a critical role when you add or subtract polynomials, or need to solve for specific values.