Problem 3 - Algebra 1: Concepts and Skills

Question

Identify the polynomial by degree and by the number of terms. $$ -9 y+5 $$

Step-by-Step Solution

Answer

Based on the solution steps, the degree of this polynomial is 1 and it has 2 terms.

Step 1: Determining the Degree of the Polynomial

The given polynomial is \(-9 y+5\). There is no exponent attached with the variable 'y', which means, by default, it carries an exponent of 1. Therefore, the degree of this polynomial is 1.

Step 2: Determining the Number of Terms in the Polynomial

The polynomial \(-9 y+5\) is made up of two parts or pieces that are separated by a '+'. These two pieces are '-9y' and '5'. Therefore, the number of terms in this polynomial is 2.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept in understanding its structure and behavior. It is defined as the largest exponent of the variable present in the polynomial. Knowing the degree helps us predict the shape of the polynomial's graph and its end-behavior.

In the given polynomial \(-9y + 5\), we observe that the variable \(y\) is present and does not carry any explicit exponent. When a variable has no visible exponent, it means it has an exponent of 1 by default. Therefore, the term \(-9y\) implies that the exponent of \(y\) is 1, which makes the degree of this polynomial 1.

To easily determine the degree of any polynomial:

• Identify all the terms of the polynomial.
• Look for the highest exponent among these terms.
• That highest exponent is the degree of the polynomial.

Understanding the degree is crucial as it informs us that this type of polynomial will generate a straight-line graph.

Number of Terms in a Polynomial

The number of terms in a polynomial indicates how many different pieces build up the polynomial expression. Each term is usually separated by either a '+' or '-' sign. These signs signal that a separate piece, or term, exists.

For the polynomial \(-9y + 5\), we can see two distinct parts due to the presence of the '+' sign. These are \(-9y\) and \(5\). Therefore, it is clear that this polynomial comprises exactly two terms.

To determine the number of terms in any polynomial:

• Identify each distinct part that is connected by addition or subtraction operators.
• Count these distinct parts to find the total number of terms.

Getting the count of terms right is essential as it reflects the polynomial's complexity and structure.

Linear Polynomial

A linear polynomial is a special type of polynomial that is characterized by having a degree of 1. When we refer to a polynomial as 'linear', it signifies that its graph will form a straight line. This simplicity makes linear polynomials foundational in algebra and calculus.

In our given example of \(-9y + 5\), the fact that the degree is 1 confirms that it is indeed a linear polynomial. This polynomial has the general form \(ax + b\), where \(a\) and \(b\) are constants, and \(x\) is the variable. Here, \(a\) is \(-9\) and \(b\) is \(5\).

Characteristics of linear polynomials include:

• They have only two terms, typically involving a variable term and a constant term.
• The highest power of the variable is always 1.
• Their graphs are straight lines, and they can slope upwards or downwards depending on the sign and magnitude of the coefficient of the variable term.

Understanding linear polynomials is crucial since they model real-world scenarios like population growth and other linear progressions.