Problem 13 - Elementary Algebra

Question

For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term. $$ 7 y^{3}+8 $$

Step-by-Step Solution

Answer

Question: Classify the given polynomial $$7 y^{3}+8$$, determine its degree, and write the numerical coefficients of each term. Answer: The given polynomial $$7 y^{3}+8$$ is a binomial with a degree of 3. The numerical coefficients are 7 for the term \(7 y^{3}\) and 8 for the constant term 8.

Step 1: Classify the polynomial as a monomial, binomial, or trinomial

To classify the polynomial, we need to count the number of terms in it. We see that the polynomial has two terms: $$ 7 y^{3} \quad \text{ and } \quad 8 $$ Since there are two terms, the polynomial is a binomial.

Step 2: Determine the degree of the polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In our given polynomial: $$ 7 y^{3}+8 $$ The highest power of y is 3 (in the term \(7 y^{3}\)). Therefore, the degree of the polynomial is 3.

Step 3: Write the numerical coefficients of each term

A numerical coefficient is the constant multiplying a variable in a term. In our given polynomial, the numerical coefficients are: For the term \(7 y^{3}\), the numerical coefficient is 7. For the term \(8\), the numerical coefficient is 8 (since it is a constant term and does not have a variable, we consider the entire term as the numerical coefficient). In conclusion, the given polynomial $$7 y^{3}+8$$ is a binomial with a degree of 3. The numerical coefficients are 7 for the term \(7 y^{3}\) and 8 for the constant term 8.

Key Concepts

Understanding a Binomial

A polynomial is an expression made up of terms, which can include variables raised to various powers and multiplied by numerical coefficients. When we talk about a binomial, we're specifically referring to a polynomial that contains exactly two terms. These terms can either be connected by a plus sign or a minus sign.
In the polynomial given in the exercise, \(7y^3 + 8\), you can easily identify it as a binomial. This is because it consists of two distinct terms: \(7y^3\) and \(8\).

• 7y^3: This is the first term with a variable \(y\).
• 8: This is the second term, a constant.

Recognizing a binomial is an important skill because it helps in understanding more complex algebraic expressions and enables us to apply appropriate algebraic techniques for simplification and calculation.

The Concept of Polynomial Degree

The degree of a polynomial is a critical concept in algebra. It tells us the highest power of the variable within the polynomial. To find the polynomial degree, look at each term and identify the exponent of its variable. The largest of these exponents is the degree of the polynomial.
For the binomial \(7y^3 + 8\), focus on the term \(7y^3\). Here, the variable \(y\) is raised to the power of 3, which is higher compared to the constant term \(8\) (remember, constants are technically \(y^0\)).

• Degree 3: The power that provides us with the degree comes from \(7y^3\), making the degree of the entire polynomial 3.

Knowing the degree of a polynomial helps determine its behavior, especially its graph's end behavior, and is crucial when performing polynomial operations like division or solving for zeros.

Numerical Coefficient Explained

The numerical coefficient in a polynomial term is the number that multiplies the variable. It's essentially the 'fixed' number in front of the variable(s). Identifying coefficients correctly is necessary for understanding and performing operations on polynomials.
In our example polynomial \(7y^3 + 8\), each term has a numerical coefficient:

• For \(7y^3\): The numerical coefficient is 7. This coefficient multiplies \(y^3\).
• For the term 8: It's a constant, so the coefficient is considered as 8 because it stands alone.

Clearly defining and understanding numerical coefficients allows you to tackle operations such as polynomial addition, subtraction, and factorization effectively. It's a foundational skill for more advanced algebra topics.