Problem 12 - Elementary and Intermediate Algebra

Question

Make a term-coefficient-degree table like that shown in Example 1 for each polynomial. $$ y^{4}-y^{3}+16 y^{2}+3 y $$ (TABLE NOT COPY)

Step-by-Step Solution

Answer

The table lists terms with coefficients 1, -1, 16, 3 and degrees 4, 3, 2, 1 respectively.

Step 1: List the Terms in the Polynomial

The given polynomial is \( y^{4} - y^{3} + 16y^{2} + 3y \). Identify each term in the polynomial:1. \( y^4 \)2. \( -y^3 \)3. \( 16y^2 \)4. \( 3y \)

Step 2: Identify the Coefficient of Each Term

For each term listed in Step 1, determine the coefficient:1. Term \( y^4 \): Coefficient is 1.2. Term \( -y^3 \): Coefficient is -1.3. Term \( 16y^2 \): Coefficient is 16.4. Term \( 3y \): Coefficient is 3.

Step 3: Determine the Degree of Each Term

The degree of a term is the exponent of the variable \( y \) in that term:1. Term \( y^4 \): Degree is 4.2. Term \( -y^3 \): Degree is 3.3. Term \( 16y^2 \): Degree is 2.4. Term \( 3y \): Degree is 1.

Step 4: Create the Term-Coefficient-Degree Table

Construct the table using the information from Steps 1-3:| Term | Coefficient | Degree ||----------|-------------|--------|| \( y^4 \) | 1 | 4 || \( -y^3 \) | -1 | 3 || \( 16y^2 \)| 16 | 2 || \( 3y \) | 3 | 1 |

Key Concepts

Understanding Polynomial Terms

A polynomial is essentially a sum of distinct parts known as terms. These terms are expressions that differ in the powers of the variables they contain. In our example of the polynomial \( y^4-y^3+16y^2+3y \), we can identify each term separately:

• \( y^4 \)
• \( -y^3 \)
• \( 16y^2 \)
• \( 3y \)

Each of these elements plays a significant role. A term is composed of a coefficient and a variable raised to an exponent. In this context, the powers of the variable \( y \) help determine how each term should be organized within the polynomial. Recognizing and breaking down each term is the first step in analyzing polynomials effectively.

Decoding Coefficients

The coefficient in a polynomial term is the numerical factor that multiplies the variable part of the term. Understanding coefficients is crucial, as they determine the strength or magnitude of each term in the polynomial. In simple terms, they tell us 'how much' of the corresponding variable exists.
For our polynomial example, consider the coefficients:

• The term \( y^4 \) has a coefficient of 1.
• The term \( -y^3 \) has a coefficient of -1, indicating a negative contribution.
• The term \( 16y^2 \) has a coefficient of 16, showing a larger influence on the polynomial's value.
• The term \( 3y \) has a coefficient of 3.

Each coefficient works alongside its respective variable and exponent to define its specific term's impact within the full polynomial.

Exploring the Degree of a Polynomial

The degree of a polynomial is determined by the highest exponent in its terms. It signifies the polynomial's overall complexity and the potential number of times a polynomial function crosses the x-axis on a graph. In each individual term, the degree refers to the exponent of the variable.
Examining our polynomial example, we can see:

• The term \( y^4 \) is of degree 4.
• The term \( -y^3 \) is of degree 3.
• The term \( 16y^2 \) is of degree 2.
• The term \( 3y \) is of degree 1.

Understanding the degree of each term helps in organizing a polynomial and in predicting its behavior across different scenarios, such as graph plotting and solving equations. Recognizing the degree also aids in connecting the polynomial to other mathematical areas such as calculus and linear algebra.