Problem 13 - 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. $$-5 x^{11}$$

Step-by-Step Solution

Answer

The expression is a monomial polynomial with degree 11.

Step 1: Identify if the Expression is a Polynomial

A polynomial is an algebraic expression made up of terms, where each term includes a variable raised to a non-negative integer power, multiplied by a coefficient. The expression given is \(-5x^{11}\). This fits the criteria for a polynomial because it consists of a single term: a constant (-5) multiplying a variable \(x\) raised to a positive integer power (11). Thus, it is a polynomial.

Step 2: Determine the Degree of the Polynomial

The degree of a polynomial is determined by the highest power of the variable \(x\) present in the expression. In the expression \(-5x^{11}\), the highest power is 11. Therefore, the degree of this polynomial is 11.

Step 3: Classify the Polynomial by the Number of Terms

A polynomial can be classified as a monomial, binomial, trinomial, or none of these based on the number of terms it contains. A monomial has one term, a binomial has two terms, and a trinomial has three terms. The expression \(-5x^{11}\) has only one term, so it is classified as a monomial.

Key Concepts

Monomial

A monomial is a type of algebraic expression that consists of exactly one term. Each monomial is made up of two parts: a coefficient and variables each raised to a non-negative integer power.
For example, in the expression \( -5x^{11} \), -5 is the coefficient and \(x^{11} \) is the variable part.
Here, the exponent is 11, which means this monomial is a power of 11.
Monomials are quite straightforward because they lack addition or subtraction operators connecting different terms.

• A monomial such as \( 4xy^{2} \) has the coefficient 4 and the variables are \( x \) and \( y^{2} \).
• Similarly, \( z^{7} \) is a monomial since it consists of a single term - just the variable \( z \) raised to the 7th power.

Monomials can have multiple variables like in \( 3abc \), yet they remain a single entity as long as they are not separated by + or - signs.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable present in the expression.
It is a crucial element that gives us insight into the function's growth and behavior.
For example, in the expression \(-5x^{11}\), the degree is 11 because \(x\) is raised to the 11th power.

The degree provides useful information:

• Simplifies Comparison: The degree helps in comparing different polynomials, particularly when determining their growth rates and end behavior.
• Indicates Root Count: A polynomial of degree n can have up to n roots (values that make the polynomial equal to zero).
• Function Analysis: Knowing the degree can help identify potential maxima and minima.

The degree is always based on the variable with the highest exponent. If you're dealing with multi-variable polynomials, check each term's degree separately and select the highest value.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions form the basis of algebra and come in various forms, from simple ones like monomials to complex arrangements like polynomials.

• Parts of an Expression: Basic components include coefficients (numerical factors), variables (letters representing numbers), and exponents (indicate repeated multiplication).
• Operations: Common operations used include addition, subtraction, multiplication, and sometimes division (except by a variable).

Expressions like \(3xy + 2y^{2} - 5\) are formed by combining multiple terms using these operations.
Algebraic expressions differ from equations as they do not include equality signs, meaning they do not assert equivalence but simply represent a value.
They serve various functions, from solving tests and problems to representing real-world situations mathematically.