Problem 19 - 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. $$-9 r^{3} s^{2}-r^{2} s^{2}+\frac{1}{2} r s+6 s$$

Step-by-Step Solution

Answer

The polynomial has four terms: \(-9 r^{3} s^{2}\), \(-r^{2} s^{2}\), \(\frac{1}{2} r s\), and \(6 s\). Their coefficients are -9, -1, \(\frac{1}{2}\), and 6, respectively. The degrees of r and s are (3, 2), (2, 2), (1, 1), and (0, 1) for each term. The degree of the polynomial is the highest total degree of its terms, which is 5.

Step 1: Identify Each Term in the Polynomial

First, we need to identify each term in the given polynomial. A term is a product of factors, and in this case, a term has the form \(a \cdot r^b \cdot s^c\), where \(a\) is a coefficient and \(r\) and \(s\) are indeterminates raised to the power of \(b\) and \(c\), respectively. The given polynomial consists of the following terms: \(-9 r^{3} s^{2}, -r^{2} s^{2}, \frac{1}{2} r s, 6 s\) Now let's move on to finding the coefficient and degree of each term.

Step 2: Identify the Coefficient and Degree of Each Term

1. For term \(-9 r^{3} s^{2}\): - The coefficient is -9 - The degree of r is 3, and the degree of s is 2. 2. For term \(-r^{2} s^{2}\): - The coefficient is -1 (as there is no number specified, it is assumed to be -1) - The degree of r is 2, and the degree of s is 2. 3. For term \(\frac{1}{2} r s\): - The coefficient is \(\frac{1}{2}\) - The degree of r is 1, and the degree of s is 1. 4. For term \(6 s\): - The coefficient is 6 - The degree of r is 0 (as there is no r term), and the degree of s is 1.

Step 3: Identify the Degree of the Polynomial

The degree of a polynomial is the highest total power of its variables in any term. In our case, we need to find the term with the highest sum of the degrees of r and s. Let's compare these in each term: 1. \(-9 r^{3} s^{2}\): The degree of r = 3, degree of s = 2, so the total degree = 3 + 2 = 5 2. \(-r^{2} s^{2}\): The degree of r = 2, degree of s = 2, so the total degree = 2 + 2 = 4 3. \(\frac{1}{2} r s\): The degree of r = 1, degree of s = 1, so the total degree = 1 + 1 = 2 4. \(6 s\): The degree of r = 0, degree of s = 1, so the total degree = 0 + 1 = 1 Now that we have compared the total degrees of each term in the polynomial, we can deduce that the degree of the polynomial is the highest total degree, which is 5, as we found for the first term (\(-9 r^{3} s^{2}\)).

Key Concepts

Term Identification

In polynomials, terms are the individual components that are added or subtracted. A term typically comprises numbers and variables combined through multiplication. For example, in the polynomial \[-9 r^{3} s^{2} - r^{2} s^{2} + \frac{1}{2} r s + 6 s\], terms are separated by plus (+) or minus (−) signs.

• Term 1: \(-9 r^{3} s^{2}\)
• Term 2: \(- r^{2} s^{2}\)
• Term 3: \(\frac{1}{2} r s\)
• Term 4: \(6 s\)

Identifying terms correctly is essential as each term contributes uniquely to the overall polynomial's characteristics. Understanding how they combine and align with each other is the basis for grasping polynomials.

Coefficient

The coefficient in a term is the numerical factor that multiplies the variable(s). Think of it as the number that scales the whole term. Knowing coefficients is essential in evaluating polynomials and understanding their behavior.

• For \(-9 r^{3} s^{2}\), the coefficient is \(-9\).
• For \(- r^{2} s^{2}\), the coefficient is \(-1\). When no number is explicitly shown in front of a variable, it is assumed to be \(-1\) or \(1\) based on the sign.
• For \(\frac{1}{2} r s\), the coefficient is \(\frac{1}{2}\).
• For \(6 s\), the coefficient is \(6\).

Spotting these coefficients helps in adding, subtracting, or even graphing polynomials, as they show the influence of each term based on different values of variables.

Degree of a Term

The degree of a term in a polynomial refers to the sum of the exponents of its variables. This gives insight into how the term behaves as the variables change. Higher degrees can indicate more rapid increases or decreases as variables increase.

• In \(-9 r^{3} s^{2}\), the degree is calculated by adding the exponents: \(3 + 2 = 5\).
• In \(- r^{2} s^{2}\), the degree is \(2 + 2 = 4\).
• In \(\frac{1}{2} r s\), it is \(1 + 1 = 2\).
• In \(6 s\), since \(r\) has no variable exponent, it's \(0 + 1 = 1\).

Identifying the degree of each term is crucial for predicting how each term will act, especially in terms of growth and dominance within the polynomial as a whole.

Degree of a Polynomial

The degree of a polynomial is determined by the term with the highest degree among all individual terms. It essentially dictates the polynomial's most characteristic behavior, mainly how it will expand or contract with variable changes.

• For our example polynomial \(-9 r^{3} s^{2} - r^{2} s^{2} + \frac{1}{2} r s + 6 s\), assess each term's degree.
• The highest degree is \(5\), from the term \(-9 r^{3} s^{2}\).

This degree gives the most weight in terms of the polynomial's maximum expansion and influences solutions, roots, and intersections in applied scenarios.