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Chapter 13: Problem 8

Find the degree of each polynomial. $$d^{2}+c^{4}$$

Short Answer

Expert verified
The polynomial has a degree of 4.

Step by step solution

01

Polynomial Identification

Identify the terms in the polynomial provided. The polynomial given is \(d^{2}+c^{4}\), which consists of two terms: \(d^{2}\) and \(c^{4}\).
02

Degree of Individual Terms

Determine the degree of each term by looking at the exponent of the variable in each term. The term \(d^{2}\) has a degree of 2, and the term \(c^{4}\) has a degree of 4.
03

Determining the Degree of the Polynomial

The degree of a polynomial is the highest degree among its terms. Compare the degrees of the individual terms (2 and 4). The highest degree is 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Terms
In mathematics, a polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. Each separate part of a polynomial is called a term.
For example, in the polynomial expression \(d^2 + c^4\), we have two distinct terms: \(d^2\) and \(c^4\).

  • The term \(d^2\) includes the variable \(d\), raised to the second power (or squared), and is considered a single term.
  • Similarly, \(c^4\) includes the variable \(c\), raised to the fourth power, and also constitutes one term of the polynomial.
Each term in a polynomial can vary considerably based on the variables, their coefficients, and the operations connecting them.
Exponents in Polynomials
The exponent in a polynomial term refers to the power to which a variable within that term is raised. It plays a crucial role in defining the term's degree. For instance, in each term of \(d^2 + c^4\):
  • The exponent of \(d\) in the term \(d^2\) is 2, meaning the variable \(d\) is multiplied by itself once.

  • On the other hand, the exponent of \(c\) in the term \(c^4\) is 4, meaning the variable \(c\) is multiplied by itself three times.
Understanding exponents is vital for calculating the degree of polynomial expressions, as they tell us how many times a base is utilized in a multiplication.
Highest Degree in Polynomials
The degree of a polynomial is a key descriptor of the polynomial's overall structure. It is determined by identifying the term with the highest exponent on its variable. In our example \(d^2 + c^4\):
  • We have two terms with degrees 2 and 4, respectively.

  • The highest degree among these terms is 4, corresponding to the term \(c^4\).
The degree of a polynomial will dictate its graph's end behavior and provide insight into the number of turning points the graph may have. Recognizing the highest degree can help students easily categorize and understand various polynomials and their properties.

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