Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, find the degree and leading coefficient for the given polynomial. $$ -2 x^{2}-3 x^{5}+x-6 $$

Step-by-Step Solution

Answer

Degree: 5, Leading Coefficient: -3.

Step 1: Identify the Polynomial

The given polynomial is \[ -2x^2 - 3x^5 + x - 6. \] We are tasked with finding the degree of this polynomial and its leading coefficient.

Step 2: Find the Degree of the Polynomial

The degree of a polynomial is determined by the term with the highest power of the variable. In this polynomial, the term \(-3x^5\) has the highest power, which is \(5\). Thus, the degree of the polynomial is \(5\).

Step 3: Identify the Leading Term

The leading term of a polynomial is the term with the highest power of the variable. Here, it is \(-3x^5\) because it contains the highest exponent.

Step 4: Find the Leading Coefficient

The leading coefficient is the coefficient of the leading term. Thus, for the term \(-3x^5\), the coefficient is \(-3\). Therefore, the leading coefficient of the polynomial is \(-3\).

Key Concepts

Leading Coefficient

When dealing with polynomials, the leading coefficient is a crucial element to understand. It is simply the coefficient of the term with the highest degree in the polynomial. In the polynomial given in this exercise, \(-3x^5 + -2x^2 + x - 6\), the highest degree is 5, found in the term \(-3x^5\).
The leading coefficient is therefore \(-3\), as it is the coefficient attached to \(x^5\).
This coefficient plays a significant role in the behavior of the graph of the polynomial, especially regarding the polynomial's end behavior. A positive leading coefficient will make the graph rise to the right, while a negative one will make it fall.
By recognizing and understanding the leading coefficient, you can gain insight into how the polynomial behaves as its inputs become very large either in the positive or negative direction. It forms a foundation for the analysis of the entire polynomial structure.

Polynomial Terms

Polynomials are expressions made up of terms that are added or subtracted from one another. Each term in a polynomial consists of a coefficient and a variable raised to an exponent. In the polynomial \(-2x^2 - 3x^5 + x - 6\), terms include \(-3x^5\), \(-2x^2\), \(x\), and \(-6\).
Each term has its own degree which is determined by the exponent of the variable within that term.

• The term \(-3x^5\) has a degree of 5.
• The term \(-2x^2\) has a degree of 2.
• The term \(x\) is equivalent to \(1x^1\) and has a degree of 1.
• The constant term \(-6\) has a degree of 0 since it does not have any variable attached to it.

The degree of the polynomial itself is determined by the term with the highest degree, which in this case is 5, from the term \(-3x^5\).
Understanding these terms, their degrees, and how they interact is essential in breaking down and analyzing the overall behavior of a polynomial.

Coefficient Identification

Coefficients are the numerical factors in a polynomial term. They are the numbers that multiply the variables or powers of variables within the terms. In working with polynomials, it is important to accurately identify the coefficients as they have a direct impact on the shape and position of the graph.
In the polynomial \(-2x^2 - 3x^5 + x - 6\), the coefficients are:

• \(-3\) for \(-3x^5\)
• \(-2\) for \(-2x^2\)
• \(1\) for \(x\), since it can be seen as \(1x\)
• \(-6\) as a constant term, although it doesn't have a variable, it's still a coefficient.

Being able to correctly identify and utilize coefficients allows you to manipulate and understand polynomials more clearly. These coefficients, along with their respective terms, help shape the entire polynomial and are key in both algebraic manipulations and graphing the function.