Problem 24 - Algebra 1: Concepts and Skills

Question

Write the polynomial in standard form. Then identify the polynomial by degree and by the number of terms. $$ 2 x $$

Step-by-Step Solution

Answer

The given expression \(2x\) is already in standard form. It is a monomial of degree 1.

Step 1: Write the polynomial in standard form

The given polynomial \(2x\) is already in the standard form. The standard form of a polynomial arranges the terms by degree in descending order. In this case, since we only have one term, it is in its standard form.

Step 2: Identify the polynomial by degree

The degree of a polynomial is the highest exponent in the polynomial. The degree of the given monomial \(2x\) is 1 because the exponent of \(x\) is \(1\) (note that if no exponent is shown, it is understood to be \(1\)).

Step 3: Identify the number of terms in the polynomial

A term is a single mathematical number, variable, or a combination of both. In our polynomial \(2x\), we only have one term, which is \(2x\). Therefore, this polynomial is a monomial since it consists of only one term.

Key Concepts

Standard Form

The standard form of a polynomial is an important concept to master when working with polynomials. It refers to the way a polynomial is structured or arranged. Specifically, a polynomial is in standard form when all of its terms are arranged in order of decreasing degree (or exponent). This means you write the highest power of the variable first, then the next highest, and so on, down to the constant term (if there is one).
For example, the polynomial \(3x^2 + 2x + 5\) is in standard form because the term with the highest degree, \(3x^2\), comes first, followed by \(2x\) and then the constant \(5\).

In the case of a single-term polynomial like \(2x\), it's already in standard form. Since there's only one term, there are no other terms with higher or lower degrees to consider.

• Always use descending order of exponents
• Start with the highest degree term
• Include all terms even if their coefficients are zero

Adhering to this format helps in simplifying the computations and comparisons between polynomials.

Degree of a Polynomial

The degree of a polynomial is a fundamental characteristic that tells you the highest power of the variable present in the polynomial. It's crucial because it gives insights into the polynomial's behavior and graph.
For example, the polynomial \(2x^3 + x^2\) has a degree of 3 because the highest exponent of the variable \(x\) is 3.

In the simplest case, like that of the monomial \(2x\), the degree is 1, as the highest (and only) exponent on the variable \(x\) is 1.

• The degree of a constant term (like \(5\)) is 0
• Degree indicates the order of the polynomial
• A higher degree often results in more complex graphs

Knowing the degree helps in identifying the polynomial and can significantly optimize solving equations involving polynomials.

Monomial

A monomial is a type of polynomial, perhaps the simplest kind. By definition, a monomial consists of only one term. This term can be a number, a variable, or a product of numbers and variables. An example is \(7x^3\), which is a monomial because it has only one term.
Monomials are the building blocks for more complex polynomials. They can be manipulated to form binomials (two terms), trinomials (three terms), and polynomials with even more terms.

In the given exercise, \(2x\) is a monomial. It has just one term, containing a constant multiplied by a variable.

• Monomials can include coefficients (numerical factors)
• Their exponents must be non-negative integers
• They are integral for understanding polynomial operations

Understanding monomials is essential, as they serve as the simplest unit in polynomial algebra, paving the way for more complex expressions and operations.