Problem 11 - Algebra Form and Function

Question

Find the degree. $$ 2 s^{6}-3 s^{5}-6 s^{4}-4 s+1 $$

Step-by-Step Solution

Answer

Answer: The degree of the given polynomial is 6.

Step 1: Identify the terms

First, let's identify each term in the polynomial: \(2s^{6}\), \(-3s^{5}\), \(-6s^{4}\), \(-4s\), and \(1\).

Step 2: Find the powers of s

Now let's list the powers of the variable s for each term: \(6\) for \(2s^{6}\), \(5\) for \(-3s^{5}\), \(4\) for \(-6s^{4}\), and \(1\) for \(-4s\). We don't consider the term \(1\) as it doesn't contain any variable and hence its power is \(0\).

Step 3: Determine the highest power

In our list of powers, the highest power of "s" is \(6\), as seen in the term \(2s^{6}\).

Step 4: Write the degree of the polynomial

Since the highest power of "s" in the polynomial is \(6\), the degree of the given polynomial is \(\boxed{6}\).

Key Concepts

Polynomial Terms

In algebra, polynomials are expressions comprising variables, coefficients, and the operations of addition, subtraction, and multiplication. These expressions are formed by several individual components known as "terms." Each term consists of a coefficient (a number), a variable (such as "s"), and an exponent, showing the number of times the variable is multiplied by itself. For example, in the polynomial \(2s^6 - 3s^5 - 6s^4 - 4s + 1\), the terms are:

• \(2s^6\)
• \(-3s^5\)
• \(-6s^4\)
• \(-4s\)
• \(1\)

Each term plays a distinct role in forming the overall structure of the polynomial. Understanding the composition of these terms is crucial for performing algebraic operations and identifying the characteristics of a polynomial.

Variable Powers

The variable power in a polynomial term refers to the exponent attached to the variable, indicating the degree to which the variable is raised. This part plays a vital role in defining the term's structure. Each term in the polynomial has its unique power:

• The term \(2s^6\) has a power of 6.
• The term \(-3s^5\) has a power of 5.
• The term \(-6s^4\) has a power of 4.
• The term \(-4s\) is actually \(-4s^1\) with a power of 1.
• The constant term \(1\) can be considered as \(1s^0\), where the variable power is 0.

Each variable's power contributes to understanding the behavior and the possible "shape" of a graph when the polynomial is plotted. Variable powers determine many of the mathematical properties of a polynomial.

Highest Power in Polynomial

In any polynomial, the highest power of a variable present among the terms is particularly significant. This highest power determines the polynomial’s degree, which provides insight into the polynomial's characteristics, such as the number of roots and the general shape of its graph.
In the provided example \(2s^6 - 3s^5 - 6s^4 - 4s + 1\), the highest power is 6, found in the term \(2s^6\).

• This indicates it is a 6th-degree polynomial.

Recognizing the highest power simplifies tasks like graphing the polynomial or predicting end behavior, which assist in understanding complex algebraic relationships more effectively.

Algebra Concepts

Algebra is a branch of mathematics dealing with symbols (like variables) and the rules for manipulating these symbols. It enables the representation and solving of equations systematically. Understanding algebraic concepts is crucial for tackling polynomials and their various properties.
Some key algebra concepts applicable to polynomials include:

• Simplification: Combining like terms and reducing expressions to simpler forms.
• Substitution: Evaluating polynomials for given variable values.
• Factoring: Expressing a polynomial as a product of simpler polynomials.
• Root Finding: Determining values of the variable that make the polynomial equal to zero.

An understanding of these algebraic concepts provides the foundation necessary for grasping more advanced topics within and beyond polynomials, enhancing problem-solving skills and mathematical reasoning.