Problem 13 - College Algebra

Question

Find the degree and leading coefficient for the given polynomial. $$7-2 x^{2}$$

Step-by-Step Solution

Answer

Degree: 2; Leading coefficient: -2.

Step 1: Identify the Polynomial Terms

The given polynomial is \(7 - 2x^2\). This polynomial has two terms: a constant term \(7\) and a term \(-2x^2\) which involves the variable \(x\).

Step 2: Determine the Degree of Each Term

The degree of a term in a polynomial is determined by the exponent of the variable in that term. For the term \(7\), which is a constant with no variable, its degree is \(0\). For the term \(-2x^2\), the degree is \(2\) as the exponent of \(x\) is \(2\).

Step 3: Find the Degree of the Polynomial

The degree of a polynomial is the highest degree of its terms when the polynomial is expressed as a sum of non-zero terms. From the terms identified, the term with the highest degree is \(-2x^2\), which has a degree of \(2\). Thus, the degree of the polynomial \(7 - 2x^2\) is \(2\).

Step 4: Identify the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree. In the polynomial \(7 - 2x^2\), the term with the highest degree is \(-2x^2\), and the coefficient of this term is \(-2\). Therefore, the leading coefficient is \(-2\).

Key Concepts

Leading Coefficient

In a polynomial, the leading coefficient is a crucial aspect in understanding its behavior. The leading coefficient is the number in front of the term with the highest degree. For example, in the polynomial \(7 - 2x^2\), the term with the highest degree is \(-2x^2\). The coefficient here is \(-2\), making it the leading coefficient.

The leading coefficient is not just any coefficient; it has significant implications for the graph of the polynomial. This coefficient can affect the width and direction (upwards or downwards) of a parabola when graphed. For instance, if the leading coefficient is positive, the graph opens upwards, while a negative leading coefficient means it opens downward. Here are a few more aspects to consider:

• A leading coefficient of zero usually indicates that the polynomial lacks terms with variables and power. It effectively becomes a constant.
• It determines the end behavior of a polynomial function's graph.

Polynomial Terms

Understanding polynomial terms is fundamental. Each term in a polynomial is a product of a constant and a variable raised to a power. These are the building blocks of your polynomial expression. Take the example of the polynomial \(7 - 2x^2\).

This polynomial consists of two parts:

• Constant term: This is the term without any variables, like the \(7\) in our example. It is simply a number and its degree is always zero.
• Variable term: Here, it is \(-2x^2\). This term includes the variable \(x\) and an exponent.

Each term in a polynomial is separated by a plus or minus sign. Understanding the structure and types of terms helps in simplification and evaluation. Adjusting coefficients, variables and powers can change the entire polynomial's behavior and solutions, making it essential to accurately identify each term.

Exponent of a Variable

The exponent in a term, more formally known as the power, denotes how many times a variable is multiplied by itself. In the polynomial \(7 - 2x^2\), the exponent is the power of \(x\) in each non-constant term.

Here, it is important to note:

• Constant term: A term like \(7\), with no visible variable, has an implied variable of \(x^0\), since any number to the power of zero is 1.
• Exponent: In \(-2x^2\), the number \(2\) is the exponent, highlighting that \(x\) is squared.

The role of the exponent is powerful in shaping the polynomial. It not only helps determine the degree of each term but also gives insight into the polynomial's structure. The highest exponent found in a polynomial determines its degree, indicating how the output value behaves concerning changes in the input. Thus, recognizing and understanding exponents are critical in solving and working with polynomial expressions.