Problem 9 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$-12$$

Step-by-Step Solution

Answer

The degree of a constant polynomial \(-12\) is 0.

Step 1: Understanding Polynomials

A polynomial can be a term or a sum of multiple terms, where each term is a product of a constant and variable(s) raised to whole number powers. In this case, \(-12\) is considered a constant polynomial.

Step 2: Identifying Constant Polynomials

Constant polynomials have only one term, which is the constant itself. This means they have no variables involved. Constant polynomials have a degree of zero by definition because there are no variable terms to indicate any power.

Step 3: Determining the Degree

For any constant polynomial, like \(-12\), since it doesn’t include a variable raised to a power, its degree is zero. The degree of a polynomial is the highest power of the variable in the polynomial, and here it is \(-\), as there are no variables.

Key Concepts

Constant Polynomials

A constant polynomial is one of the simplest forms of polynomials. It contains only a single term which is a constant value. Variables are not involved in constant polynomials, making them unique.

• Unlike other polynomials, a constant polynomial does not change, no matter what value is substituted for the variable.
• Constant polynomials can be positive, negative, or even zero, like \(5\), \(-12\), or \(0\).
• The term consists solely of a number without any variable part.

This makes constant polynomials easy to work with since there is no dependency on a variable. Whether you think about it as a horizontal line on a graph or a fixed number, its simplicity is what defines it.

Degree of a Polynomial

The degree of a polynomial is a key characteristic that tells you the highest power of the variable present. For more complex polynomials, it gives an idea of how the function behaves as you move along the graph.

• In typical polynomials, the degree is determined by the term with the largest exponent.
• For example, the polynomial \(3x^2 + 5x + 1\) has a degree of 2 because the highest exponent on the variable x is 2.
• Constant polynomials, however, don't have a variable or exponent, which means their degree is defined as zero.

This helps in understanding the roots or solutions of the equation, as well as predicting the shape of the graph associated with the polynomial expression. Recognizing the degree lets you classify and identify polynomials efficiently.

Variable Terms

Variable terms are components of a polynomial that involve variables (often denoted by letters like \(x\) or \(y\)) raised to a power.

• Each variable term consists of a coefficient (the constant part) and a variable part.
• For example, in the term \(7x^3\), "7" is the coefficient and "\(x^3\)" is the variable part.
• Variable terms drive the behavior of polynomials since they introduce variations based on different inputs.

Understanding variable terms is crucial when analyzing polynomials since they determine the degree and significantly influence the polynomial's graph. Without them, you are left with constant polynomials, which are much more limited in dynamics and potential behaviors.