Question

Write each polynomial in descending order of degree. $$2 n^{4}-5 n^{3}-n+2+n^{6}-7 n^{8}$$

Short answer

Question: Arrange the terms of the polynomial \(-7n^8 - 5n^3 + 2n^4 + n^6 - n + 2\) in descending order of degree. Answer: \(-7n^8 + n^6 + 2n^4 - 5n^3 - n + 2\)

Step-by-step solution

Identify the terms with the respective degree

List the terms of the polynomial along with their degrees: - \(-7n^8\) with degree 8 - \(n^6\) with degree 6 - \(2n^4\) with degree 4 - \(-5n^3\) with degree 3 - \(-n\) with degree 1 - \(2\) with degree 0 (constant term)

Arrange the terms in descending order of degree

Rewrite the polynomial by placing the terms in order of degree from highest to lowest: $$-7n^8 + n^6 + 2n^4 - 5n^3 - n + 2$$

Simplify (if necessary)

In this case, there are no like terms to simplify in the polynomial. Thus, the final polynomial in descending order of degree is: $$-7n^8 + n^6 + 2n^4 - 5n^3 - n + 2$$

Key concepts

Polynomial Degrees

Understanding the degrees of a polynomial is crucial when learning algebra. A polynomial is a mathematical expression consisting of variables and coefficients that come together in a series of terms. Each term is comprised of a variable raised to an exponent, and this exponent is what we call the degree of the term. For example, in the term -7n^8, the exponent 8 indicates the degree of that term.

When referring to the degree of the entire polynomial, we look for the term with the highest exponent. This term dictates the overall degree of the polynomial, signifying its highest power. In our example, the original polynomial's degree is determined by the term with -7n^8, making it a polynomial of degree 8. This concept is fundamental because the degree of a polynomial can give us a lot of information about the equation's behavior, like its possible number of roots and its end behavior.

Arranging Polynomials

Arranging polynomials in descending order of degree helps to organize and streamline the process of simplifying and solving, as well as makes it easier to understand the polynomial's structure and behavior. Starting with the highest degree term and ending with the constant term (degree 0), we create an orderly progression from the most dominant term to the least.

The procedure involves first identifying each term's degree as we've done in step 1 of our solution. Once identified, we then rewrite the polynomial, placing terms in descending order by their degree, shown in step 2. Here's how to view the process:

• Term with highest degree goes first.
• Follow with terms of lesser degrees.
• End with the constant term, if there is one.

This order not only keeps our work neat but also adheres to the conventional presentation of polynomials, facilitating easier communication and comprehension of algebraic work.

Algebraic Expressions

Algebraic expressions are the backbone of algebra and encompass polynomials along with other types of expressions. They consist of numbers, variables, and operators (like +, -, * and /) combined to show the value of something. For simplicity and clarity, we often perform operations like addition and subtraction to combine like terms, which haven't occurred in the example polynomial as no two terms share the same degree.

An effective strategy includes:

• Identify like terms (terms with the same variable raised to the same power).
• Combine these terms by addition or subtraction.
• Keep the expression arranged from higher to lower degrees, or vice versa if the context requires.

Such strategies not only make algebraic expressions more accessible to understand but also pave the way for more advanced operations like differentiation and integration in calculus.