Problem 17 - Beginning and Intermediate Algebra

Question

For each polynomial, identify each term in the polynomial, the coefficient and degree of each term, and the degree of the polynomial. $$7 y^{3}+10 y^{2}-y+2$$

Step-by-Step Solution

Answer

In the polynomial \(7 y^{3}+10 y^{2}-y+2\), we have four terms: \(7y^{3}\), \(10y^{2}\), \(-y\), and \(2\). Their coefficients are \(7\), \(10\), \(-1\), and \(2\), respectively. Their degrees are \(3\), \(2\), \(1\), and \(0\), respectively. The degree of the polynomial is \(3\), which is the highest degree among its terms.

Step 1: Identify each term

First, we need to identify each term in the polynomial. A term is a product of a coefficient and a variable raised to a power. In the given polynomial \( 7 y^{3}+10 y^{2}-y+2 \), we have 4 terms: \(7y^{3}\), \(10y^{2}\), \(-y\), and \(2\).

Step 2: Identify the coefficient of each term

The coefficient of a term is the number in front of a variable (or constant term). Here are the coefficients of each term in the polynomial: - For the term \(7y^{3}\), the coefficient is \(7\). - For the term \(10y^{2}\), the coefficient is \(10\). - For the term \(-y\), the coefficient is \(-1\). Note that when there is no number in front of the variable, it means the coefficient is \(1\) or \(-1\) depending on the sign. - For the constant term \(2\), the coefficient is simply the constant value itself, which is \(2\).

Step 3: Identify the degree of each term

The degree of a term is the exponent of the variable in the term. Here are the degrees of each term in the polynomial: - For the term \(7y^{3}\), the degree is \(3\). - For the term \(10y^{2}\), the degree is \(2\). - For the term \(-y\), the degree is \(1\). Note that when there is no exponent, it's understood that the exponent is \(1\). - For the constant term \(2\), the degree is \(0\). Constant terms always have a degree of \(0\), as they do not have a variable in them.

Step 4: Identify the degree of the polynomial

The degree of a polynomial is the highest degree among all its terms. In the given polynomial \( 7 y^{3}+10 y^{2}-y+2 \), we see the highest degree is \(3\). Thus, the degree of the polynomial is \(3\).

Key Concepts

Coefficients

In mathematics, a coefficient is a number that is multiplied by the variable in a term of a polynomial. This number provides the term with its magnitude, or how much it contributes to the overall polynomial. For example, in the polynomial \( 7y^3 + 10y^2 - y + 2 \), the coefficients are:

• 7 in the term \( 7y^3 \)
• 10 in the term \( 10y^2 \)
• -1 in the term \( -y \) \(\text{ (remember that if no number is written, it means it is 1 or the sign if negative)}\)
• 2 in the constant term \( 2 \)

Each coefficient tells us how many times that term's value should be counted. Recognizing coefficients helps in solving polynomials and understanding their structure.

Degree of a Polynomial

The degree of a polynomial is a crucial concept in algebra. It refers to the highest power of the variable in the polynomial. This helps us determine how the graph of the polynomial looks and how it behaves as the value of the variable changes. In the polynomial \( 7y^3 + 10y^2 - y + 2 \), the degrees of the individual terms are:

• The term \( 7y^3 \) has a degree of 3.
• The term \( 10y^2 \) has a degree of 2.
• The term \( -y \) has a degree of 1.
• The constant term \( 2 \), has a degree of 0.

To find the degree of the polynomial itself, we simply look for the term with the highest degree, which in this case is 3 from \( 7y^3 \). Therefore, the degree of the polynomial is 3. Understanding the degree of a polynomial is important because it indicates the polynomial's overall level of complexity.

Terms of a Polynomial

A polynomial is made up of terms, and each term is a distinct part of the polynomial's overall expression. Every term can include constants, variables, and exponents, but keeps its own identity.Let's break down the polynomial \( 7y^3 + 10y^2 - y + 2 \) into its individual terms:

• \( 7y^3 \)
• \( 10y^2 \)
• \( -y \)
• \( 2 \)

In any polynomial:

• Each term is separated by a plus (+) or minus (-) sign.
• The number of terms can affect the name given to the polynomial, for example, a polynomial with two terms is known as a binomial, whereas one with three terms is a trinomial.

Identifying the individual terms in a polynomial is essential for performing algebraic operations such as addition, subtraction, multiplication, and factoring. Each term's individual characteristics, such as its coefficients and degrees, help us further in understanding the polynomial's behavior.