Problem 13

Question

For the following exercises, find the degree and leading coefficient for the given polynomial. $$ 7-2 x^{2} $$

Step-by-Step Solution

Verified

Answer

The degree is 2 and the leading coefficient is -2.

1Step 1: Identify the terms of the polynomial

The given polynomial is $7 - 2x^2$. It consists of two terms: the constant term $7$ and the term $-2x^2$.

2Step 2: Determine the degree of the polynomial

The degree of a polynomial is the highest power of the variable $x$ in any of the terms. In $7 - 2x^2$, the term with the highest power of $x$ is $-2x^2$, where the power of $x$ is 2. Therefore, the degree of the polynomial is 2.

3Step 3: Identify the leading term of the polynomial

The leading term of a polynomial is the term with the highest degree. For $7 - 2x^2$, the leading term is $-2x^2$ because it has the highest degree of 2.

4Step 4: Find the leading coefficient

The leading coefficient is the coefficient of the leading term. In the term $-2x^2$, the coefficient is $-2$. Therefore, the leading coefficient of the given polynomial is $-2$.

Key Concepts

Leading CoefficientTerms of a PolynomialConstant Term

Leading Coefficient

A key element of any polynomial, the leading coefficient is the number that sits in front of the term with the highest degree. Think of it as the polynomial's front-runner. In our example, the polynomial is $7 - 2x^2$.

To find the leading term, look for the term with the highest power of $x$.
This term is $-2x^2$, where the exponent on $x$ is 2.
The coefficient of this term, $-2$, is the leading coefficient.

The leading coefficient gives important information about how the polynomial behaves, especially for large values of $x$. If it's negative, like in our case, the polynomial will drop as $x$ increases. When you spot a leading coefficient, you get a sneak peek into the shape of the polynomial's graph.

Terms of a Polynomial

Polynomials are like mathematical sentences made up of terms, each consisting of numbers and variables. These terms are separated by addition or subtraction signs. For the polynomial $7 - 2x^2$, there are two terms:

$7$ - This is known as the constant term because it's just a number without a variable attached.
$-2x^2$ - This is a term that includes both a coefficient, $-2$, and a variable, $x$, raised to an exponent, $2$.

Each term contributes to the overall expression, but they play different roles based on their variables and coefficients. Understanding these terms is crucial for identifying key properties like the degree and the leading coefficient.

Constant Term

In any polynomial, the constant term is the one without a variable. It's simply a standalone number. In the polynomial $7 - 2x^2$, the constant term is $7$.

Since it doesn't change, no matter what $x$ is, it stays constant as the name suggests.
The constant term is crucial when evaluating the polynomial at $x = 0$; it's exactly what you'll find the entire polynomial equal to.

Constant terms are not influenced by the variable $x$ and often provide the starting point on the graph, indicating where the polynomial crosses the $y$-axis. Recognizing the constant term helps us understand the initial value of a polynomial before any other terms have a significant effect.

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