Problem 15 - A Graphical Approach to College Algebra

Question

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. $$18 p^{6} q+6 p q$$

Step-by-Step Solution

Answer

The expression is a polynomial, it is a binomial of degree 7.

Step 1: Determine if the expression is a polynomial

To identify if the expression \(18 p^{6} q + 6 p q\) is a polynomial, check that each term has non-negative integer exponents. Both terms, \(18 p^{6} q\) and \(6 p q\), have variables raised to positive integers, which complies with the definition of a polynomial. Therefore, this expression is a polynomial.

Step 2: Identify the degree of the polynomial

The degree of a polynomial is the highest sum of the exponents of the variables in a single term. The first term \(18 p^{6} q\) has exponents \(p^6\) and \(q^1\), summing to \(6 + 1 = 7\). The second term \(6 p q\) has exponents \(p^1\) and \(q^1\), summing to \(1 + 1 = 2\). Since \(7\) is the highest sum, the degree of the polynomial is \(7\).

Step 3: Classify the polynomial by the number of terms

The polynomial \(18 p^{6} q + 6 p q\) has two distinct terms. A polynomial with two distinct terms is classified as a binomial.

Key Concepts

Understanding the Degree of a Polynomial

In mathematics, when dealing with polynomials, the degree of a polynomial is a critical concept. It tells us the highest power of its variable. To find it, we look at the terms of the polynomial and consider the exponents of the variables within each term.

• A term in a polynomial is the product of a number (coefficient) and variables raised to an exponent.
• The degree of a polynomial is the highest sum of the exponents for a single term in the polynomial.
• The example polynomial is \(18 p^{6} q + 6 p q\).
• For the term \(18 p^{6} q\), the exponents are 6 and 1, so the total is \(6+1=7\).
• For the term \(6 p q\), the exponents are both 1, so the total is \(1+1=2\).
• The polynomial's degree is thus the highest sum, which is \(7\).

Understanding the degree helps predict the behavior of the polynomial. For instance, it indicates the polynomial's end behavior on a graph.

Monomial, Binomial, and Trinomial Explained

Polynomials classify based on the number of terms they include: monomials, binomials, and trinomials are the main types. Understanding these classifications helps in organizing and simplifying mathematical expressions.

• Monomial: A polynomial with just one term. For example, \(3x^2\) is a monomial.
• Binomial: A polynomial with exactly two terms, such as \(x + 5\).
• Trinomial: A polynomial with exactly three terms, such as \(x^2 + 3x + 2\).
• Our example, \(18 p^{6} q + 6 p q\), has two terms, so it is a binomial.

These classifications help in identifying the form of polynomial operations, such as addition and subtraction.

Non-Negative Integer Exponents in Polynomials

A requisite for an expression to be a polynomial is that it must have non-negative integer exponents. This means all exponents should be whole numbers and greater than or equal to zero.

• Exponents indicate how many times a variable multiplies itself.
• In \(18 p^{6} q + 6 p q\):
  • The exponents \(p^{6}\) and \(q^{1}\) are non-negative integers.
  • This qualifies each term as a true polynomial term.
• Ensuring all exponents are non-negative integers confirms the validity of a polynomial.

Expressions with variable exponents as fractions or negatives don't qualify as polynomials. Identifying non-negative integer exponents is crucial for the correct interpretation of polynomial forms.