Problem 2 - Algebra 2

Question

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(2 b+4 b^{3}-3 b^{5}-7\)

Step-by-Step Solution

Answer

Degree is 5 and leading coefficient is -3.

Step 1: Identify the Polynomial Form

The given polynomial is \(2b + 4b^3 - 3b^5 - 7\). It consists of terms that involve the variable \(b\) raised to non-negative integer powers. All terms are expressed with respect to a single variable \(b\), hence it's a polynomial in one variable.

Step 2: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in any of the terms. Looking at the terms \(2b = b^1\), \(4b^3\), and \(-3b^5\), the highest power is \(5\). Therefore, the degree of this polynomial is \(5\) as it corresponds to the term \(-3b^5\).

Step 3: Identify the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree. In the polynomial \(2b + 4b^3 - 3b^5 - 7\), the term with the highest degree is \(-3b^5\). The coefficient of this term is \(-3\). Hence, the leading coefficient is \(-3\).

Key Concepts

Degree of a Polynomial

When we talk about the "degree of a polynomial," we're referring to the highest power of the variable within the polynomial. In simpler terms, it tells us the most prominent exponent you can find in any term. For example, consider the polynomial given in the exercise: \[2b + 4b^3 - 3b^5 - 7\] To determine its degree, inspect each term:

• First term: \(2b = b^1\) – here, the power is 1.
• Second term: \(4b^3\) – the power is 3.
• Third term: \(-3b^5\) – the power is 5.
• Last term: \(-7\) – as a constant, this term's power is 0.

Among these terms, \(-3b^5\) has the highest power, which is 5. Therefore, the degree of this polynomial is 5. This degree helps us understand how the polynomial behaves as the variable grows larger, giving us insight into the "size" or complexity of the polynomial.

Leading Coefficient

In a polynomial, the "leading coefficient" is the coefficient of the term with the highest degree. The coefficient is the number that directly multiplies the variable. Let's take another look at the polynomial: \[2b + 4b^3 - 3b^5 - 7\] The term \(-3b^5\) has the highest degree, which means its coefficient, \(-3\), stands as the leading coefficient. Understanding the leading coefficient is essential since it plays a significant role in determining the polynomial's end behavior on a graph. If the leading coefficient is positive, the polynomial ends up going to infinity as the variable increases. If it's negative, like in our case, the polynomial tends to fall towards negative infinity as the variable grows.

Single Variable Polynomial

A "single variable polynomial" consists of terms that feature only one variable raised to non-negative integer powers. It's a simple, yet foundational concept in algebra. Consider the polynomial provided: \[2b + 4b^3 - 3b^5 - 7\] Notice how each term involves only the variable \(b\):

• Term \(2b\) has \(b^1\)
• Term \(4b^3\) has \(b^3\)
• Term \(-3b^5\) has \(b^5\)
• Constant term \(-7\) involves \(b^0\) or no variable power, essentially.

All components align to one variable \(b\), highlighting that this is indeed a polynomial in a single variable. This simplicity makes them easier to work with, as you're only dealing with one dimensionality—perfect for building your understanding of more complex polynomials involving multiple variables later on.