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Chapter 10: Problem 9

In the following exercises, determine the degree of each polynomial. $$8 a^{5}-2 a^{3}+1$$

Short Answer

Expert verified
The degree of the polynomial is 5.

Step by step solution

01

Identify Each Term

Identify each term in the polynomial: 1. 8a^5 2. -2a^3 3. 1
02

Find the Degree of Each Term

Determine the degree of each term by looking at the exponent of the variable 'a' in each term: 1. The term 8a^5 has a degree of 5.2. The term -2a^3 has a degree of 3.3. The term 1 is a constant term with a degree of 0.
03

Determine the Highest Degree

Compare the degrees of all the terms identified: 1. 8a^5 has a degree of 5. 2. -2a^3 has a degree of 3. 3. The constant term has a degree of 0. The highest degree among these terms is 5.
04

Conclude the Degree of the Polynomial

The degree of a polynomial is the highest degree of its terms. Therefore, the degree of the polynomial 8a^5 - 2a^3 + 1 is 5.

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Key Concepts

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

Polynomial Terms
A polynomial is made up of terms, which are the building blocks of the expression. Terms are separated by plus and minus signs. In the polynomial from the exercise, we have three terms: 8a^5, -2a^3, and 1.

Each term consists of a coefficient (a number), a variable (like 'a'), and an exponent (power of the variable).

For example, in the term 8a^5:
  • 8 is the coefficient
  • a is the variable
  • 5 is the exponent
Understanding the composition of polynomial terms is essential in determining the degree and solving other problems.

In summary, ensure you can identify and understand each term's components to analyze and work with polynomials effectively.
Exponents
Exponents are a crucial part of polynomial terms. They show how many times a variable is multiplied by itself. In the term 8a^5, the exponent is 5, indicating that 'a' is multiplied by itself five times: a * a * a * a * a.

When you analyze a polynomial, look for the exponents of the variables in each term. This helps determine the degree of each term.

Here are key things to remember about exponents:
  • Exponents are written as small numbers to the right and above the variable.
  • If there is no exponent shown, like in 'a', assume the exponent is 1.
  • For constant terms, the exponent of the variable can be considered as 0.
Understanding exponents is essential for finding the degree of a polynomial because the degree is determined by the term with the highest exponent.
Constant Term
The constant term in a polynomial is a term that does not contain any variables. In our exercise, the constant term is '1'.

When determining the degree of a polynomial, constant terms have an exponent of 0, because any number times itself zero times is 1.

Here are some key points about constant terms:
  • They remain the same regardless of the value of the variable.
  • Their exponent is always zero.
  • They affect the value of the polynomial but not its degree.
Remember, the presence of a constant term is important for the overall value of the polynomial but not for determining its degree. This concept can simplify solving and analyzing polynomial problems.

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Most popular questions from this chapter

Simplify. $$\left(\frac{3}{5}\right)^{2}$$

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Simplify. $$\left(m^{5}\right)^{-1}$$

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Simplify each expression. $$(s+8)(s+9)$$
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