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Task 2: The System of Polynomials
You are aware of the different types of numbers: natural numbers, integers, rational numbers,
and real numbers. Now you will work with a property of the number system called the closure
property. A set of numbers is closed for a specific mathematical operation if you can perform
the operation on any two elements in the set and always get a result that is an element of the
set.
2
Consider the set of natural numbers. When you add two natural numbers, you will always get
a natural number. For example, 3 + 4 = 7. So, the set of natural numbers is said to be closed
under the operation of addition,
Similarly, adding two integers or two rational numbers or two real numbers always produces
an integer, or rational number, or a real number, respectively. So, all the systems of numbers
are closed under the operation of addition,
a. Think of polynomials as a system. For each of the following operations, determine
whether the system is closed under the operation. In each case, explain why it is closed
or provide an example showing that it isn't.
1. Addition
Type your response here:

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Sagot :

devishri1977

Answer:

Step-by-step explanation:

a) 1) Polynomial is closed under addition. In polynomial addition, only the coefficient change and variables and it powers does not change.

Example: Add 3x² - 2x + 7  and -2x² - 5x + 3

3x² - 2x + 7 - 2x² - 5x + 3 = 3x² -2x² - 2x - 5x + 7 + 3

= x² - 7x + 10

2)Polynomial    is closed under addition. In polynomial subtraction, only the coefficient change and variables and it powers does not change.

Example: Subtract 3x - 2 from 4x + 2

4x + 2 - (3x - 2) = 4x + 2 - 3x + 2  = 7x + 4

3) Polynomial is closed under multiplication. In this, exponents of the variables are added according to the exponent rule. As the exponent of polynomials are whole numbers and whole numbers are closed under addition, the result will be a whole number.

-2x³ * 3x² = [tex]-3*x^{3+2}=-3x^{5}[/tex]

4) Polynomial is not closed under multiplication. In division, we may get a negative exponent. So, it is not closed under division.

[tex]\dfrac{x^{4}}{x^{9}}=x^{4-9}=x^{-5}\\\\x^{-5} \ is \ not \ a \ polynomial.[/tex]

                                 Addition        subtraction  multiplication   Division

Natural numbers          yes             not closed    Yes                   Not closed

Integers                        Yes                 Yes              Yes                   not closed

Rational numbers         Yes                 Yes              Yes                     not closed

Irrational numbers       not closed   not closed   not closed   not closed

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