Question

Determine the leading term of \(3+2 x-7 x^{3}\).

Short answer

The leading term is -7x^3.

Step-by-step solution

Identify all terms in the polynomial

The given polynomial is: a = 3 + 2x - 7x^3Identify each term within the polynomial.- The constant term: 3- The linear term: 2x- The cubic term: -7x^3

Determine the degree of each term

The degree of each term is defined by its exponent on the variable x:- The constant term: 3 has degree 0.- The linear term: 2x has degree 1.- The cubic term: -7x^3 has degree 3.

Find the highest degree term

To determine the leading term, identify the term with the highest degree:- The constant term: 3 (degree 0)- The linear term: 2x (degree 1)- The cubic term: -7x^3 (degree 3)The term with the highest degree (3) is -7x^3.

Key concepts

Polynomial Terms

In a polynomial, the expression is made up of terms. A term is a product of a constant and variables raised to exponents. For example, in the polynomial given, we have three terms:

• The constant term: This is just a number without any variables. In our polynomial, it is 3.
• The linear term: This term includes a variable raised to the power of 1. In our example, it is 2x.
• The cubic term: This term includes a variable raised to the power of 3. Here, it is -7x^3.

Recognizing the different terms in a polynomial is the first step towards understanding its structure and behavior.

Degree of Polynomial

The degree of a polynomial is the highest exponent found on one of its terms. It indicates the highest power that the variable is raised to within the polynomial. To find the degree:

• First identify each term in the polynomial.
• Next, determine the exponent of the variable in each term. For example, in 2x, the exponent is 1 since it is equivalent to 2x^1.
• The constant term like 3 can be thought of as 3x^0 since any number to the power of zero is 1. Hence, the degree here is 0.
• The cubic term -7x^3 has an exponent of 3.

So, in our polynomial 3 + 2x - 7x^3, the degrees are 0, 1, and 3 respectively. The degree of the polynomial itself is the highest degree of its terms, which is 3 in this case.

Cubic Term

The cubic term in a polynomial is the term where the variable is raised to the power of 3. For our example, this term is -7x^3. Here are some key points about cubic terms:

• They are essential in defining the behavior of a polynomial, especially for higher degree polynomials.
• The coefficient in the cubic term (-7 in this case) significantly affects the shape and direction of the polynomial graph.
• Cubic terms indicate that the polynomial will have inflection points, where the curve changes direction.

To determine the leading term, you need to find the term with the highest degree, which in this equation is the cubic term -7x^3.

Linear Term

In a polynomial, the linear term is the term where the variable is raised to the power of 1. In our given polynomial, the linear term is 2x. Here’s what to remember about linear terms:

• They represent the linearly changing part of the polynomial.
• The coefficient (2 in this case) indicates the slope if you were to graph just this term.
• Linear terms are fundamental in understanding the changes in the polynomial's value at different points.

While the linear term is crucial, the leading term is determined by finding the term with the highest degree, and in this specific polynomial, it's not the linear term but the cubic term -7x^3 since 3 is greater than 1.