Problem 65 - Algebra A Combined Function

Question

Write each polynomial in descending powers of the variable and with no missing powers. See Example 15. $$ 8 y+2 y^{4} $$

Step-by-Step Solution

Answer

The polynomial is \( 2y^4 + 8y \).

Step 1: Identify the terms

The given polynomial is \( 8y + 2y^4 \). It consists of two terms: \( 8y \) and \( 2y^4 \).

Step 2: Arrange terms by power

We need to rearrange the polynomial so that the terms are in order from highest power to lowest power of \( y \). Identify the powers of each term: \( 2y^4 \) is degree 4, and \( 8y \) is degree 1. Thus, \( 2y^4 \) comes first.

Step 3: Write the polynomial

Write the polynomial in descending order of the powers of \( y \): \( 2y^4 + 8y \). This is the polynomial arranged in the required format.

Key Concepts

Descending Powers

When organizing polynomials, arranging them in descending powers means ordering the terms by the exponent of the variable, from largest to smallest. This helps in quickly identifying the highest degree of the polynomial and simplifies operations like addition and subtraction.

• Identify the variable's power in each term. In the example polynomial, we have \(8y\) and \(2y^4\).
• Rank these terms by their exponent values. Here, \(2y^4\) has a higher power (4) compared to \(8y\), which is only 1.
• Rearrange the terms to start with the largest power: \(2y^4 + 8y\).

This way, anyone looking at the polynomial can immediately see which term will dominate its behavior as the variable grows larger.

Variable Terms

Polyomials consist of multiple terms, each made up of a coefficient (a number) and a variable raised to a power. In the given polynomial \(8y + 2y^4\), each term is dependent on the variable \(y\).

• The term \(8y\): Here, \(8\) is the coefficient, and \(y\) is the variable raised to the power of 1.
• The term \(2y^4\): The coefficient is \(2\), and \(y\) is raised to the power of 4.

Understanding variable terms helps in manipulating and simplifying polynomial expressions accurately. They form the building blocks of the polynomial and directly affect its degree and structure.

Polynomial Degree

The degree of a polynomial is determined by the term with the highest power of the variable. It plays a crucial role in understanding the polynomial's properties and behavior.
- To find the degree, look at each term in a polynomial.- In \(8y + 2y^4\), the term \(2y^4\) has the highest power, which is 4.- Thus, the degree of the polynomial is 4.Knowing the degree is essential for anticipating how the polynomial behaves as \(y\) increases or decreases, as it gives insight into the polynomial's growth or decay rate.

Rearranging Terms

Rearranging terms within a polynomial involves sorting each term according to the power of its variable, prioritizing from highest to lowest.

• Begin by listing out the terms: \(8y\) and \(2y^4\).
• Order the terms by descending power: since \(2y^4\) has a power of 4 and \(8y\) has a power of 1, \(2y^4\) should come first.
• Rewriting the polynomial in the correct order yields \(2y^4 + 8y\).

This rearrangement not only adheres to mathematical standards but also makes further operations such as integration or differentiation more straightforward.