Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the algebraic expression is a polynomial. If it is, write the polynomial in standard form and state its degree.\(w^{2}-w^{4}+2 w^{3}\)

Step-by-Step Solution

Answer

The given expression \(w^{2}-w^{4}+2 w^{3}\) is a polynomial. In its standard form, it becomes \(-w^{4}+2 w^{3}+w^{2}\) , and its degree is 4.

Step 1: Identifying the Algebraic Expression

First, scrutinize the given algebraic expression \(w^{2}-w^{4}+2 w^{3}\). It has three terms: \(w^{2}\), \(-w^{4}\) and \(2 w^{3}\). All the terms are of the form \(ax^n\), which is the structure of a polynomial term.

Step 2: Arranging in Standard Form

The standard form of a polynomial organizes its terms in descending order of their variable exponents. Taking our given algebraic expression, we can rearrange it as \(-w^{4}+ 2 w^{3} + w^{2}\). This, consequently, is the standard form of the given algebraic expression.

Step 3: Determining the Degree

The degree of a polynomial is determined by the highest exponent in the expression. In the standard form of our polynomial, the highest exponent is 4, hence, the degree of the polynomial is 4.

Key Concepts

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and mathematical operations. It's like a language composed of terms, where each term consists of a coefficient (a number) and a variable raised to an exponent. These expressions can represent real-world quantities and are foundational to algebra.
Keep in mind these points when identifying algebraic expressions:

• They can have one or more terms.
• Terms are usually separated by addition or subtraction.
• Variables can be any letter.

In our exercise, we encountered the expression \(w^{2} - w^{4} + 2w^{3}\). This expression features terms connected predominantly through addition or subtraction: \(w^2\), \(-w^4\), and \(2w^3\). Each term reflects the general format of an algebraic expression. Understanding how these terms connect and contribute to the entirety of an expression is crucial.

Polynomial Degree

The degree of a polynomial is one of its essential attributes, as it tells us about the expression's complexity. The degree is defined as the highest power of the variable in the polynomial. This concept holds for any polynomial, no matter how many terms it contains.
When determining polynomial degree:

• Look for the largest exponent in the expression.
• The largest exponent tells us the polynomial's degree.

In our example, the polynomial \(-w^{4} + 2w^{3} + w^{2}\) features terms with exponents of 4, 3, and 2. The term \(-w^4\) possesses the greatest exponent—4—thus establishing the polynomial's degree as 4. The degree helps you to categorize and compare polynomials.

Standard Form of Polynomial

A polynomial's standard form provides a structured way to present an expression. It arranges terms in descending order based on the exponent of the variable. Writing polynomials in their standard form makes them easier to read and analyze, especially when comparing them to other polynomials.
To write a polynomial in standard form:

• Identify all terms by their exponents.
• Reorder them starting from the highest to the lowest exponent.

Consider our polynomial \(w^{2} - w^{4} + 2w^{3}\). After reorganizing it, the standard form is \(-w^{4} + 2w^{3} + w^{2}\). This reorganization doesn't change the polynomial's value; it simply orders the expression logically, facilitating easier computation and comparison. Using this form is a fundamental practice in algebra, providing clarity to polynomial expressions across different contexts.