Problem 165 - Precalculus

Question

For the following exercises, 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 Terms

The polynomial given is \( 7 - 2x^2 \). It has two terms: \( 7 \) and \( -2x^2 \).

Step 2: Determine the Degree

The degree of a polynomial is the highest power of the variable \( x \) in the polynomial. In \( 7 - 2x^2 \), the highest power of \( x \) is 2, found in the term \( -2x^2 \). Therefore, the degree of the polynomial is 2.

Step 3: Find the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power. In \( 7 - 2x^2 \), the term with the highest power is \( -2x^2 \), and its coefficient is \(-2\). Thus, the leading coefficient is \(-2\).

Key Concepts

Degree of Polynomial

The degree of a polynomial is a critical term that refers to the highest power of the variable within the polynomial. It is a measure of the polynomial's highest exponent. For instance, in the polynomial \(7 - 2x^2\), the exponent that is the most significant is 2, located in the term \(-2x^2\). Thus, the degree is 2. Understanding the degree is fundamental because it gives you insights into the polynomial's behavior, such as its shape when graphed and the number of potential solutions for equations set equal to zero. The degree can also influence the end behavior of the polynomial.

Leading Coefficient

The leading coefficient is another important aspect of a polynomial. It is the coefficient of the term with the highest degree. This coefficient significantly impacts the shape and characteristics of the polynomial function. For the polynomial \(7 - 2x^2\), the term with the highest degree is \(-2x^2\). The coefficient here is \(-2\), making it the leading coefficient.

• Determining the leading coefficient is crucial, especially when graphing polynomials, as it affects the direction and steepness of the graph.
• A positive leading coefficient implies the ends of the graph will trend upwards, while a negative indicates they will point downward.
• This insight helps predict the overall behavior of polynomial functions.

Terms of Polynomial

Polynomials are mathematical expressions composed of terms. Each term is a combination of constants, coefficients, and variables raised to a power. In the context of our example polynomial, \(7 - 2x^2\), the terms are \(7\) and \(-2x^2\). These terms are separated by addition or subtraction operators.

• Constant term: In this polynomial, the constant term is 7. This is significant because it remains constant no matter the value of \(x\).
• Variable term: The term \(-2x^2\) involves the variable raised to a power, specifically \(x^2\). The term's degree is determined by the highest power of the variable, which is 2 here.
• Each term contributes to the polynomial's overall form and how it behaves graphically.

Understanding the structure of polynomials by dissecting their terms gives you a deeper insight into their properties and applications.