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Chapter 13: Problem 25

Find the degree of each polynomial. $$x^{2}+3 x+2$$

Short Answer

Expert verified
The degree of the polynomial \(x^{2}+3x+2\) is 2.

Step by step solution

01

Identify Each Term

A polynomial is composed of terms, each having a degree defined by the power of its variable. Analyze the given polynomial: \[ x^{2} + 3x + 2 \] There are three terms in this polynomial: \(x^2\), \(3x\), and \(2\).
02

Determine the Degree of Each Term

Now, examine the degree of each term individually:- The first term \(x^2\) has a degree of 2.- The second term \(3x\) has a degree of 1, since \(3x\) is equivalent to \(3x^1\).- The third term \(2\) is a constant, which has a degree of 0.
03

Identify the Highest Degree

In a polynomial, the degree is defined as the highest degree of its terms:- From Step 2, the degrees are: 2 (from \(x^2\)), 1 (from \(3x\)), and 0 (from \(2\)).- The highest degree among these terms is 2, which is the degree of the polynomial.

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Key Concepts

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

Understanding Polynomial Terms
A polynomial is an expression made up of several terms. Each term can have a variable raised to an exponent, coupled with a coefficient. In any given polynomial, like in the example \(x^2 + 3x + 2\), recognizing the individual terms is crucial. Here:
  • The term \(x^2\) consists of the variable \(x\) raised to the power of 2. It doesn't have an explicit coefficient, which implies a coefficient of 1.
  • The term \(3x\) combines the coefficient 3 with the variable \(x\), where the exponent is 1.
  • The term \(2\) is a constant because it doesn鈥檛 involve a variable; it stands alone without any accompaniments.
Identifying these terms helps you in further understanding their roles in forming the complete polynomial expression.
Defining the Degree of Polynomials
The degree of a polynomial is a critical concept, as it indicates the highest power of the variable in the polynomial. It tells us a lot about the polynomial's behavior and structure. Let鈥檚 explore this concept further:
  • Each term in a polynomial has its own degree, which is dictated by the exponent of the variable. For instance, the term \(x^2\) has a degree of 2, while \(3x\) is of degree 1.
  • To find the degree of the entire polynomial, we search for the term with the highest degree. In the polynomial \(x^2 + 3x + 2\), the term \(x^2\) holds the highest degree value, which is 2.
This highest degree signifies the polynomial鈥檚 degree, which in this example is 2. Understanding the degree helps in predicting how a polynomial behaves, especially in graphing.
Clarifying the Constant Term in Polynomials
A constant term in a polynomial is a term that has no variable attached to it. In essence, its role is essential yet straightforward in polynomial expressions:
  • The constant term is always a number by itself. In \(x^2 + 3x + 2\), the constant term is "2." It doesn't change regardless of the values the other terms take.
  • By definition, the degree of a constant term is 0. This influences the polynomial, especially when visualizing it in equations or graphs.
  • Constant terms add to the overall constant value in arithmetic operations of the polynomial, providing an important reference point.
Recognizing and understanding constant terms is important in simplifying and evaluating polynomials effectively.

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