Problem 39 - Elementary and Intermediate Algebra

Question

Find the degree of each polynomial. See Example \(1 .\) $$ -5 r^{2} s^{2}-r^{3} s+3 $$

Step-by-Step Solution

Answer

The degree of the polynomial is 4.

Step 1: Identify the Terms

The polynomial given is \[ -5r^2s^2 - r^3s + 3 \]It consists of three terms: 1. \(-5r^2s^2\), 2. \(-r^3s\), and 3. \(+3\).

Step 2: Determine Degree of Each Term

For each term, the degree is the sum of the exponents of the variables within that term.- For term \(-5r^2s^2\), the degree is \[2 + 2 = 4\]- For term \(-r^3s\), the degree is \[3 + 1 = 4\]- For the constant term \(+3\), the degree is \[0\] because it has no variable.

Step 3: Find the Highest Degree

The degree of the polynomial is the highest degree of any individual term.From the terms \(-5r^2s^2\) and \(-r^3s\), both have the highest degree, which is \[4\].

Key Concepts

Polynomial Terms

In the realm of mathematics, polynomials are expressions made up of terms separated by plus or minus signs. Each term consists of numbers and variables multiplied together. For instance, in the polynomial - \(-5r^2s^2 - r^3s + 3\), we see it has three separate terms:

• The first term, \(-5r^2s^2\), includes two variables \(r\) and \(s\) with their respective exponents and a coefficient of \(-5\).
  
• The second term, \(-r^3s\), has a single variable \(r\) with an exponent and another variable \(s\). The coefficient here is \(-1\) although it is not explicitly written.
  
• The last term, \(+3\), is a constant term, consisting solely of a number without any variables.
  

Understanding the makeup of each polynomial term is essential for performing operations such as finding the degree of the polynomial or expanding and factoring polynomials.

Polynomial Degree

The degree of a polynomial is a concept that indicates the highest power of the variable(s) within the polynomial. It helps determine the behavior and shape of the polynomial graph.
To find the degree, focus on each term in the polynomial and sum the exponents of the variables of each term:

• For the term \(-5r^2s^2\), the exponents for \(r\) and \(s\) are both 2. Adding them together, the degree is \(2 + 2 = 4\).
  
• In \(-r^3s\), the exponent for \(r\) is 3 and for \(s\) it is 1. The degree of this term is \(3 + 1 = 4\).
  
• For the constant term \(+3\), since there are no variables, the degree is \(0\).
  

Once the degrees of all terms are calculated, the degree of the polynomial is the highest degree of its terms. Thus, the ultimate degree of the polynomial \(-5r^2s^2 - r^3s + 3\) is 4.

Exponents in Polynomials

Exponents play a crucial role in polynomials as they define the power to which a variable is raised in any given term. Understanding how to interpret exponents is key in determining not just the degree of a polynomial, but also managing polynomial operations.
In terminology, the exponent tells us how many times the base - usually a variable - is multiplied by itself. For example, in the term \(-5r^2s^2\), both \(r\) and \(s\) have exponents of 2, thereby suggesting each variable is squared.
Similarly, the term \(-r^3s\) has the variable \(r\) raised to the power of 3, indicating \(r\) is utilized as a factor three times, while \(s\) is used once, implicitly having an exponent of 1.
By correctly interpreting exponents, you can not only determine the degree of individual terms but also understand the overall dynamics and complexity within the given polynomial.