Problem 20 - Beginning and Intermediate Algebra

Question

For each polynomial, identify each term in the polynomial, the coefficient and degree of each term, and the degree of the polynomial. $$8 m^{2} n^{2}+0.5 m^{2} n-m n+3$$

Step-by-Step Solution

Answer

The given polynomial is \(8m^{2}n^{2} + 0.5m^{2}n - mn + 3\). The identified terms are: 1. \(8m^{2}n^{2}\): Coefficient: 8, Degree: 4 2. \(0.5m^{2}n\): Coefficient: 0.5, Degree: 3 3. \(-mn\): Coefficient: -1, Degree: 2 4. \(3\): Coefficient: 3, Degree: 0 The degree of the polynomial is 4.

Step 1: Identify each term of the polynomial

To identify each term, we look for expressions that are separated by addition or subtraction signs. In this case, there are four terms in the polynomial: 1. \(8m^{2}n^{2}\) 2. \(0.5m^{2}n\) 3. \(-mn\) 4. \(3\)

Step 2: Find the coefficient and degree of each term

Now we will find the coefficient and degree of each term. 1. Term 1: \(8m^{2}n^{2}\) Coefficient: 8 Degree: The degree of this term is the sum of the powers of the variables, which is \(2 + 2 = 4\). 2. Term 2: \(0.5m^{2}n\) Coefficient: 0.5 Degree: The degree of this term is the sum of the powers of the variables, which is \(2 + 1 = 3\). 3. Term 3: \(-mn\) Coefficient: -1 (since there is no number in front of the variables, the coefficient is -1) Degree: The degree of this term is the sum of the powers of the variables, which is \(1 + 1 = 2\). 4. Term 4: \(3\) Coefficient: 3 (since it is a constant term, it only has a coefficient) Degree: The degree of a constant term is always 0.

Step 3: Find the degree of the polynomial

To find the degree of the polynomial, we must look for the term with the highest degree. In this case, the first term has the highest degree of 4. Therefore, the degree of the polynomial is 4.

Key Concepts

Polynomial Terms

A polynomial is an expression made up of terms. Each term is a product of a coefficient and variables raised to a power. To identify terms within a polynomial, look for expressions separated by addition or subtraction signs. In the example polynomial \(8m^{2}n^{2} + 0.5m^{2}n - mn + 3\), we identify the following four terms:

• \(8m^{2}n^{2}\)
• \(0.5m^{2}n\)
• \(-mn\)
• \(3\)

Each term can include coefficients and variables raised to some power, or it may be a constant term like \(3\). Understanding the structure and separation of polynomial terms is crucial when dealing with more complex algebraic expressions.

Coefficient

The coefficient of a term in a polynomial is the numerical factor that multiplies the variables within the term. It essentially tells you "how much" of that particular term exists. Let's break down the coefficients from our given polynomial:

• The coefficient of \(8m^{2}n^{2}\) is 8.
• The coefficient of \(0.5m^{2}n\) is 0.5.
• The coefficient of \(-mn\) is -1. This is because no number is visible, implying a one, and the negative sign indicates -1.
• Finally, the term \(3\) is a constant, and its own coefficient is 3.

Knowing the coefficients is essential for simplifying expressions, solving equations, and understanding the quantitative aspects of terms.

Degree of a Term

The degree of a term is the sum of the exponents of the variables in the term. It gives you an idea of how "large" the term is in terms of its variable components. Here's how to identify the degree for each term in the polynomial:

• For \(8m^{2}n^{2}\), add the powers of \(m\) and \(n\) to get \(2 + 2 = 4\).
• In \(0.5m^{2}n\), add \(2\) from \(m^{2}\) and \(1\) from \(n\) to get \(3\).
• The term \(-mn\) has \(m\) and \(n\) both raised to the first power, resulting in a degree of \(1 + 1 = 2\).
• The constant term \(3\) has no variables, and thus its degree is \(0\).

Understanding the degree of each term helps in identifying the behavior of the polynomial, particularly when graphing or comparing polynomials.

Degree of a Polynomial

The degree of a polynomial is determined by finding the term with the highest degree among all the terms. In simpler terms, you look for the "largest" term in the polynomial. Let’s see how this works in our given polynomial:

• The first term, \(8m^{2}n^{2}\), has a degree of 4, which is the highest degree in the polynomial.

Therefore, the degree of the entire polynomial is 4. This gives us valuable information about the polynomial’s behavior, particularly the shape of its graph, as it indicates the number of roots and the general form of polynomial function.