Problem 11 - Algebra A Combined Function

Question

Find the degree of each polynomial and determine whether it is a monomial, binomial, trinomial, or none of these. See Examples 2 and 3. $$ 3 z-5 z^{4} $$

Step-by-Step Solution

Answer

The polynomial is a binomial with a degree of 4.

Step 1: Identify Each Term

The given polynomial is \(3z - 5z^4\). This expression consists of two terms: \(3z\) and \(-5z^4\).

Step 2: Determine the Degree of Each Term

The degree of a term is the sum of the exponents of the variables in the term. In the term \(3z\), the exponent of \(z\) is 1, so the degree is 1. In the term \(-5z^4\), the exponent of \(z\) is 4, so the degree is 4.

Step 3: Find the Degree of the Polynomial

The degree of a polynomial is the highest degree of its terms. Between the terms \(3z\) and \(-5z^4\), the highest degree is 4. Therefore, the degree of the polynomial \(3z - 5z^4\) is 4.

Step 4: Classify the Polynomial

A polynomial is classified based on the number of terms it contains: one term is a monomial, two terms is a binomial, three terms is a trinomial, and more than three is none of these. The expression \(3z - 5z^4\) has two terms, so it is classified as a binomial.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a crucial concept because it tells us the highest power of the variable present in the polynomial. When dealing with polynomials, each term within the polynomial has its own degree. The degree of an individual term is determined by the sum of the exponents of the variables in that term.

For example, in the term \(3z\), the exponent is 1 since \(z^1 = z\). In the term \(-5z^4\), the exponent is 4 because the variable \(z\) is raised to the power of 4. To find the degree of the entire polynomial, you look for the term with the highest degree. This means the polynomial \(3z - 5z^4\) has a degree of 4 because of the term \(-5z^4\).

In practice, the degree helps us understand the behavior of the polynomial, such as the possible number of roots and the general form of its graph.

Monomial

A monomial is a polynomial with exactly one term. This single term can be a constant (like 5), a variable (like \(z\)), or a product of constants and variables (like \(3z^2\)). What makes a monomial distinct is its simplicity—it doesn't contain additions or subtractions separating different terms.

Monomials often appear as building blocks for more complex polynomials. They are straightforward and can be multiplied or added together to form binomials, trinomials, and more. When identifying a monomial, remember that it can include coefficients and multiple variables, as long as they're part of a singular product.

Some examples of monomials include:

• \(4x^3\)
• -9
• \(abc\)

Understanding monomials is foundational for recognizing and working with more complex polynomial structures.

Binomial

A binomial is a polynomial consisting of exactly two distinct terms. These terms are either added or subtracted. Because of this, binomials often serve as simple examples of polynomial expressions. The expression you worked with, \(3z - 5z^4\), is a perfect example of a binomial: it consists of the terms \(3z\) and \(-5z^4\).

The separation of terms by a plus or minus sign is what characterizes a binomial. Each term in a binomial can have its own degree based on the exponent of its variable, and the larger of these degrees will determine the degree of the entire binomial.

Binomials provide a basis for many algebraic concepts, including binomial expansions and the Binomial Theorem, both of which are instrumental in more advanced algebra. Some common examples of binomials might include:

• \(x + 7\)
• \(a^2 - 4b\)

Trinomial

A trinomial is a polynomial that has exactly three terms. Like monomials and binomials, trinomials are categorized according to the number of terms they contain. Each term in a trinomial is separated by a plus or minus sign, making them distinct from one another.

Trinomials are commonly used in algebra, especially in quadratic expressions like \(ax^2 + bx + c\). Solving trinomials often involves factoring, where the expression is rewritten as a product of simpler polynomials. Understanding how to work with trinomials is essential for solving quadratic equations and understanding their graphs.

Here are a few examples of trinomials:

• \(x^2 + 3x + 2\)
• \(4a^3 - 2a + 1\)

Each trinomial will have its degree determined by the highest degree term. These concepts will aid in classifying and working with various polynomial expressions.