Question

Identify the leading term: \(-5 x^{4}+8 x^{2}-2 x^{7}\)

Short answer

-2x^7

Step-by-step solution

- Understand the terms

Identify each term in the polynomial - The polynomial has three terms: 1. \( -5x^4 \) 2. \( 8x^2 \) 3. \( -2x^7 \)

- Identify the degree of each term

Determine the exponent for each term:1. The degree of \( -5x^4 \) is 4.2. The degree of \( 8x^2 \) is 2.3. The degree of \( -2x^7 \) is 7.

- Find the highest degree term

Compare the degrees of the terms:- The term \( -2x^7 \) has the highest degree at 7.

- Identify the leading term

The leading term is the term with the highest degree.- Therefore, the leading term is \( -2x^7 \)

Key concepts

Polynomials

A polynomial is a mathematical expression composed of variables, coefficients, and exponents. Each part of the expression separated by a plus or minus sign is called a term. For example, in the polynomial \(-5x^4 + 8x^2 - 2x^7\), we have three terms: \(-5x^4\), \(8x^2\), and \(-2x^7\). Polynomials can have any number of terms. They are used in various fields of science and engineering to model relationships and solve equations.
Keep in mind that the highest exponent in a term determines the degree of the polynomial. Understanding how to work with polynomials is crucial for many areas of mathematics, including algebra and calculus.

Degree of a Term

The degree of a term in a polynomial is the exponent of the variable in that term. Each term in a polynomial can have a different degree. For example:

• In the term \(-5x^4\), the degree is 4 because 4 is the exponent of \(x\).
• In the term \(8x^2\), the degree is 2 because 2 is the exponent of \(x\).
• In the term \(-2x^7\), the degree is 7 because 7 is the exponent of \(x\).

To compare terms in a polynomial, look at their degrees. The term with the highest degree indicates the most influence in the polynomial's behavior as the variable increases or decreases.

Highest Degree

Finding the highest degree in a polynomial is crucial because it helps identify the leading term. The leading term is the term with the highest degree and it determines the polynomial's end behavior. To find the highest degree:

• List out the terms of the polynomial.
• Identify the exponent in each term, which represents the degree.
• Compare the degrees of all terms.

In our example, \(-2x^7\) has the highest degree of 7 compared to \(-5x^4\) and \(8x^2\). Hence, \(-2x^7\) is the leading term. Understanding the leading term helps you analyze and predict the polynomial's behavior, especially as the value of the variable becomes very large or very small.