Problem 32 - 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. $$ 2 a+9 $$

Step-by-Step Solution

Answer

Answer: The polynomial is a binomial with a degree of 1. The numerical coefficients are 2 and 9.

Step 1: Classify the polynomial

The given polynomial is $$2a+9$$. It contains two terms, "\(2a\)" and "\(9\)". When a polynomial has exactly two terms, it is called a binomial. Therefore, the given polynomial is a binomial.

Step 2: Determine the degree and numerical coefficients

To find the degree of the polynomial, we identify the highest power of the variable. In this case, the only variable is "\(a\)", and its highest power is 1. Therefore, the degree of the polynomial is 1. Next, we need to identify the numerical coefficients of the terms. A numerical coefficient is the constant multiplier of a term. For the term "\(2a\)", the numerical coefficient is 2, and for the constant term "\(9\)", the numerical coefficient is 9. So in conclusion, the given polynomial is a binomial with a degree of 1. The numerical coefficients of the terms are: 2 and 9.

Key Concepts

Monomial

A monomial is a type of polynomial that has only one term. It's the simplest form of a polynomial. A term in a monomial can include numbers, variables, or the product of numbers and variables, but there are no plus or minus signs separating terms.

Here are some examples of monomials:

• 5x - A monomial because there is only one term: "5x".
• 2 - Just a single number also qualifies as a monomial.
• x^2y - A product of variables with coefficients.

Notice that a monomial can be any whole number or a power of a variable multiplied by a coefficient, but with no addition or subtraction involved.

Binomial

A binomial is a polynomial that consists of exactly two terms. These terms are separated by a plus or minus sign. Binomials are essential building blocks in algebra and are used in various operations like expansion using the binomial theorem.

Consider the polynomial in the exercise, "2a + 9", this is a classic example of a binomial because it has two terms: "2a" and "9".

• 5x - 3y - Two distinct terms separated by a minus sign.
• x^2 + 4 - Two distinct terms separated by a plus sign.

Each term leads to creative and complex expressions in mathematics, making binomials exceedingly versatile.

Trinomial

A trinomial is a type of polynomial that consists of exactly three terms. These terms can include combinations of variables with different coefficients or powers and are separated by addition or subtraction.

Some typical examples of trinomials include:

• x^2 + 3x + 2 - It has three terms with increasing powers of the variable "x".
• y + z + 1 - This contains three separate terms: "y", "z", and "1".

Trinomials often appear in quadratic equations, notably taking the form of ax^2 + bx + c, making them vital for solving algebraic equations.

Degree of Polynomial

The degree of a polynomial signifies the highest power of the variable present in the polynomial. Determining the degree is fundamental as it dictates the behavior and the graph shape of the polynomial.

For instance, in the example "2a + 9", the highest power of the variable "a" is 1, making it a polynomial of degree 1. Here is how you can identify the degree in different scenarios:

• 3x^4 + x^2 + 1 - The highest power is 4, making it degree 4.
• x^3 - 5x - The highest power is 3, so this is a degree 3 polynomial.

Understanding the degree helps in graphing polynomials and predicting their maximum number of roots or turning points.

Numerical Coefficient

A numerical coefficient is a constant multiplier of the terms in a polynomial. It provides scale to the terms, telling us how many times that term is counted.

Let's look at the polynomial "2a + 9" as covered in the exercise:

• The term "2a" has a numerical coefficient of 2.
• The constant term "9" itself serves as its numerical coefficient.

Recognizing numerical coefficients is crucial for simplifying expressions, especially in algebraic procedures like factoring or when combining like terms.