Problem 9 - 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. $$ 12 x^{4}-x^{6}-12 x^{2} $$

Step-by-Step Solution

Answer

Degree is 6; it is a trinomial.

Step 1: Identify Terms

First, let's identify the terms of the given polynomial. The polynomial is given as \( 12x^4 - x^6 - 12x^2 \). The terms are \( 12x^4 \), \( -x^6 \), and \( -12x^2 \).

Step 2: Determine Individual Degrees

Next, we determine the degree of each term. The degree of the term \( 12x^4 \) is 4 because the exponent of \( x \) is 4. The degree of \( -x^6 \) is 6, and the degree of \( -12x^2 \) is 2.

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is determined by the highest degree among its terms. Here, the degrees are 4, 6, and 2. Therefore, the degree of the polynomial is 6 since it is the highest.

Step 4: Classify the Polynomial

To classify the polynomial, we count the number of terms. This polynomial has three terms: \( 12x^4 \), \( -x^6 \), and \( -12x^2 \). Therefore, it is a trinomial because it consists of three terms.

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is essential to grasp its behavior and characteristics. The degree of a polynomial is defined as the highest power of the variable in the expression.
For instance, in the polynomial \(12x^4 - x^6 - 12x^2\), each term has a specific degree:

• \(12x^4\) has a degree of 4.
• \(-x^6\) has a degree of 6.
• \(-12x^2\) has a degree of 2.

Once you identify the degree of each term, the degree of the entire polynomial is simply the highest of these degrees. In our example, the term \(-x^6\) has the highest degree at 6. Hence, the polynomial's degree is 6. Knowing the degree enables us to predict the polynomial’s graph behavior, such as its end behavior and the number of potential turning points.

Trinomial

A polynomial can contain different numbers of terms, leading to different names. When a polynomial has three terms, it is referred to as a trinomial.
Let's look closer at the problem: \(12x^4 - x^6 - 12x^2\). This polynomial consists of three separate terms:

• \(12x^4\)
• \(-x^6\)
• \(-12x^2\)

Since there are exactly three terms, the polynomial classifies as a trinomial. Recognizing a polynomial as a monomial, binomial, or trinomial helps in applying specific algebraic techniques and simplifies understanding its structure and solving related equations.

Terms of a Polynomial

Terms in a polynomial are among its fundamental components. Each term in a polynomial is a product of numbers and variables, employing non-negative integer exponents. In the case of \(12x^4 - x^6 - 12x^2\), the terms are easily distinguishable:

• \(12x^4\) is a term with a coefficient of 12 and an exponent 4 on \(x\).
• \(-x^6\) is a term with an understood coefficient of -1 and an exponent 6 on \(x\).
• \(-12x^2\) is a term with a coefficient of -12 and an exponent 2 on \(x\).

The importance of understanding terms lies in recognizing their role in finding the degree, simplifying expressions, and performing operations like addition, subtraction, and multiplication of polynomials. Each term contributes individually to the polynomial's overall degree and influences its mathematical properties and behavior.