Problem 8 - Algebra and Trigonometry

Question

Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial, then list its terms and state its degree. \(2 x^{5}+4 x^{2}\)

Step-by-Step Solution

Answer

The polynomial is a binomial, its terms are \(2x^5\) and \(4x^2\), and its degree is 5.

Step 1: Identify the Type of Polynomial

A polynomial can be classified based on the number of terms it has. A monomial has one term, a binomial has two terms, and a trinomial has three terms. The given polynomial is \(2x^5 + 4x^2\). It has two terms: \(2x^5\) and \(4x^2\). Therefore, this polynomial is a binomial.

Step 2: List the Terms of the Polynomial

We need to list the individual terms of the polynomial. The terms in the polynomial \(2x^5 + 4x^2\) are \(2x^5\) and \(4x^2\). These terms are separated by a plus sign.

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the expression. For \(2x^5 + 4x^2\), the powers of the terms are 5 and 2. The highest power is 5, so the degree of the polynomial is 5.

Key Concepts

Degree of a Polynomial

The degree of a polynomial plays a crucial role in determining the behavior of the polynomial function. Simply put, the degree is the highest power of the variable in the polynomial's expression. For instance, consider the polynomial \(2x^5 + 4x^2\). Here, the term \(2x^5\) contains the highest exponent, which is 5. Therefore, the degree of this particular polynomial is 5.

Knowing the degree of a polynomial allows us to understand several aspects of its graph:

• The number of roots or solutions the polynomial can have, which is at most equal to its degree.
• The end behavior of the polynomial, such as how the graph acts as \(x\) approaches positive or negative infinity.
• The possible number of turning points, which can be up to one less than the degree.

Tracking the degree tells us about how the polynomial grows and helps in solving equations involving polynomials.

Monomial

A monomial is the simplest form of a polynomial, consisting of just one term. For example, an expression like \(7x^3\) or a constant like 5 can be a monomial. A term in a monomial includes a coefficient and a variable raised to a non-negative integer exponent.

Characteristics of a monomial include:

• It does not contain addition or subtraction signs within it.
• It can be just a number (called a constant monomial) or can include variables.
• The degree of a monomial is simply the exponent of its variable if it is one, or zero if the monomial is a constant.

Monomials are building blocks for larger polynomials and are often used in algebraic expressions and equations.

Binomial

A binomial consists of two distinct terms, separated by either a plus or minus sign. An example of a binomial is the expression \(2x^5 + 4x^2\) which is part of our exercise. This polynomial consists of two terms: \(2x^5\) and \(4x^2\).

Here are key features of a binomial:

• It always has exactly two terms.
• The terms can consist of constants, variables, or both. Each term may have different exponents.
• In terms of degree, the degree of the binomial is determined by the highest degree of any of its terms, just like in any other polynomial.

Understanding binomials is fundamental, as they often appear in algebraic operations involving factoring, expanding expressions using the distributive property, and solving polynomial equations.