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Chapter 7: Problem 13

In Exercises \(1-18,\) answer the following questions: (a) How many terms are there? (b) What is the degree of each term? (c) What is the degree of the polynomial? $$2 x^{3}+y^{5}$$

Short Answer

Expert verified
The polynomial has 2 terms. The degrees of the terms are 3 and 5. The highest degree is 5.

Step by step solution

01

- Identify the terms

Identify the distinct terms in the polynomial. The given polynomial is \[2x^3 + y^5\]The terms are identified as : \[2x^3 \text{ and } y^5\]
02

- Count the terms

Count how many terms are present. There are 2 terms in the polynomial: \[2x^3 \text{ and } y^5\].
03

- Find the degree of each term

Identify the power of the variable in each term. The degree of a term is the exponent of the variable: -The degree of \[2x^3\] is 3.-The degree of \[y^5\] is 5.
04

- Determine the polynomial's degree

The degree of the polynomial is the highest degree of any term in the polynomial. The terms have degrees 3 and 5.Thus, the degree of the polynomial is 5.

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Key Concepts

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

terms in polynomials
A polynomial is composed of multiple terms, which are separated by addition or subtraction. Each term is a combination of constants (numbers) and variables (like x or y) that are multiplied together. In the polynomial \[2x^3 + y^5\], there are two terms: \[2x^3 \text{ and } y^5\].
Counting terms is straightforward: look for numbers or symbols separated by + or - signs.
When you find an addition or subtraction, you鈥檝e found a boundary between terms.
In our example:
  • \(2x^3\): represents one term.
  • \(y^5\): signifies another term.

So, there are exactly two terms in \[2x^3 + y^5\].
degree of a term
The degree of a term is the highest power (exponent) of the variable in that term. To find it, identify the exponent attached to each variable.
Let鈥檚 look at the polynomial \[2x^3 + y^5\]:
  • In \[2x^3\], the variable is \(x\) and its exponent is 3. Therefore, the degree of \[2x^3\] is 3.
  • In \[y^5\], the variable is \(y\) and its exponent is 5. Thus, the degree of \[y^5\] is 5.

The degree of each term provides insight into how the term behaves as the variable grows larger. Higher degrees indicate terms that grow quickly.
degree of a polynomial
The degree of a polynomial is determined by the term with the highest degree. It represents the most significant factor in the polynomial鈥檚 growth as the variables increase.
Consider the polynomial \[2x^3 + y^5\]:
First, identify the degrees of each term:
  • Degree of \[2x^3\]: 3
  • Degree of \[y^5\]: 5

The highest of these is 5. Hence, the degree of the polynomial \[2x^3 + y^5\] is 5.
Knowing the degree helps in understanding the polynomial鈥檚 behavior and is crucial for solving polynomial equations and inequalities effectively.

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