Problem 33 - Elementary Algebra

Question

For the following problems, classify each of the polynomials as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term. $$ 4 y^{3}+3 y+1 $$

Step-by-Step Solution

Answer

Question: Classify the polynomial 4y^3 + 3y + 1, state its degree, and write the numerical coefficient of each term. Answer: The polynomial is a trinomial with a degree of 3. The numerical coefficients of the terms are 4, 3, and 1 respectively.

Step 1: Identify the number of terms

The given polynomial is: $$ 4y^3 + 3y + 1 $$ We can see that there are three terms in this polynomial: 1. \(4y^3\) 2. \(3y\) 3. \(1\) Since there are three terms, we can classify this polynomial as a trinomial.

Step 2: Identify the degree of the polynomial

The degree of a polynomial is the highest power of the variable present in the polynomial. In our given polynomial, the terms have different powers of the variable \(y\): 1. \(4y^3\): Power 3 2. \(3y\): Power 1 3. \(1\): Power 0 (since the term doesn't contain the variable \(y\)) The highest power is 3, so the degree of the polynomial is 3.

Step 3: Write the numerical coefficient of each term

The numerical coefficient is the constant factor associated with each term of the polynomial. We can easily find the numerical coefficients for each term in the given polynomial: 1. \(4y^3\): The numerical coefficient is \(4\) 2. \(3y\): The numerical coefficient is \(3\) 3. \(1\): The numerical coefficient is \(1\) To summarize, the given polynomial is a trinomial with a degree of 3. The numerical coefficients of the terms are 4, 3, and 1 respectively.

Key Concepts

Classification of Polynomials

Polynomials are generally classified based on the number of terms they contain. A **term** in a polynomial is a distinct part separated by a plus (+) or minus (-) sign. Understanding the number of terms helps us to classify them accurately.

There are three primary classifications of polynomials:

• **Monomial**: A polynomial with just one term, such as \(7x^2\) or \(-3\).
• **Binomial**: A polynomial with two terms, like \(4x - 5\). Think of a binomial as a bicycle, which stands on two wheels.
• **Trinomial**: A polynomial with three terms, for example, \(4y^3 + 3y + 1\), which is the polynomial in our exercise.

The given polynomial \(4y^3 + 3y + 1\) is classified as a trinomial because it contains exactly three terms. Recognizing these classifications helps simplify understanding and managing polynomials in mathematics.

Degree of a Polynomial

The **degree of a polynomial** is determined by the highest power (exponent) of the variable in the polynomial. In simple terms, it shows the largest exponent any term in the polynomial has. Knowing the degree is crucial as it represents the polynomial's behavior, especially at the ends of its graph and in its growth.

• For the term \(4y^3\), the power of \(y\) is 3.
• For the term \(3y\), the power of \(y\) is 1.
• The term \(1\) has a power of \(y\) equal to 0, since it doesn't involve \(y\).

In the example \(4y^3 + 3y + 1\), the highest power of the variable is 3 from the term \(4y^3\). Therefore, the degree of this polynomial is 3. This means the polynomial is cubic.

Numerical Coefficient

The **numerical coefficient** is the numerical part that multiplies the variable(s) in a term. It's like the plain number factor before any variables in a term, and it's important for evaluating and simplifying expressions. Let’s look at each term of the provided polynomial separately:

• For the term \(4y^3\), the numerical coefficient is 4.
• For the term \(3y\), the numerical coefficient is 3.
• For the term \(1\), the numerical coefficient is simply 1, as there are no variables attached.

Identifying the numerical coefficients in a polynomial can help in understanding the multiplication and scaling effects of terms within algebraic operations. It plays a fundamental role in adding, subtracting, and multiplying polynomials.