Question

State the degree of each term.$$3 x y$$

Short answer

The degree of the term '3xy' is 2.

Step-by-step solution

Identify the Terms

The given expression is a single term: '3xy'.

Understanding Degrees

The degree of a term in a polynomial is the sum of the exponents of the variables in the term.

Calculate the Degree of the Given Term

Here, the degree of the term '3xy' is the sum of the exponents of 'x' and 'y'. Since both variables are to the first power (as an exponent of '1' is implied), the degree of the term is 1 + 1 = 2.

Key concepts

Calculating Degrees

When you hear the term 'degree of a polynomial,' think of it as a way to measure the polynomial's complexity. Calculating the degree is surprisingly straightforward. Essentially, you look for the highest sum of exponents in the polynomial's terms. For a single term like in our example, \(3xy\), you just add up the exponents of all the variables. It might be tricky when the exponents are not written explicitly since, by convention, any variable without an exponent actually has an exponent of 1. So, in our case, both \(x\) and \(y\) are actually to the power of 1. The sum of these exponents, 1 + 1, gives us a degree of 2. Remember, only non-negative integer exponents are considered for calculating degrees in polynomials.

Here's how to conceptualize it: if you were to write down all the terms of a polynomial in descending order of their degree, the degree of the polynomial would be the same as the degree of the first term. This step is vital in understanding the overall 'shape' and properties of the polynomial equation.

Polynomial Terms

Polynomials are like mathematical sentences, and each term is a word in that sentence. A polynomial term is a product of constants and variables, and can look quite varied – from a simple constant like 3, to a more complex product like \(3xyz^2\). Each term is typically separated by a plus (+) or minus (-) sign in an equation. Regular engagement with different polynomial expressions will develop your ability to quickly identify and categorize terms.

In the step-by-step solution provided, \(3xy\) is identified as a single term. It's important to differentiate between terms because when we solve equations or simplify expressions, we often do so term by term. Understanding what constitutes a term also helps when you start performing operations like adding or subtracting polynomials, where like terms are combined. Remember, terms are only 'like' if they have the exactly same variable parts regardless of the coefficient.

Exponents in Polynomials

Each variable in a polynomial term can have an exponent, which is a small number located in the upper right corner next to the variable indicating how many times that variable is multiplied by itself. For instance, in the term \(x^3\), the exponent is 3, which means \(x\) is multiplied by itself three times, as in \(x \times x \times x\).

In the example \(3xy\), both \(x\) and \(y\) have an implied exponent of 1. We don’t usually write this exponent out, but it’s understood to be there. So, \(x\) is actually \(x^1\) and \(y\) is \(y^1\). This is a crucial aspect to grasp because exponents are central to operating with polynomials. They determine the power and behavior of each term within an equation. The proper handling of exponents is key when you are multiplying terms, finding derivatives, or integrating polynomial functions. Always ensure the exponents are treated with respect to their operational laws, such as the power of a product or the power of a power.