This is a full assignment I’m struggling on if someone can answer them all and get most correct I will give you Brainliest for y
our helpfulness! Thank you lots!
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:
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:
1 answer:
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 =
4) Polynomial is not closed under multiplication. In division, we may get a negative exponent. So, it is not closed under division.
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
You might be interested in
If period of 
is one-half the period of 
and
<span> has a period of 2π, then 
and 
.
</span>
To find the period of sine function
we use the rule 
.
<span /><span />
f is sine function where f (0)=0, then c=0; with period
, then 
, because 
.
To find a we consider the condition
, from where 
.
If the amplitude of is twice the amplitude of , then has a product factor twice smaller than and while period of <span> </span> is 2π and g(0)=0, we can write 
.
<span>
</span>
To find the period of sine function
<span /><span />
f is sine function where f (0)=0, then c=0; with period
To find a we consider the condition
If the amplitude of
The trapezoidal approximation will be the average of the left- and right-endpoint approximations.
Let's consider a simple example of estimating the value of a general definite integral,
Split up the interval![[a,b]](https://tex.z-dn.net/?f=%5Ba%2Cb%5D)
into 
equal subintervals,
![[x_0,x_1]\cup[x_1,x_2]\cup\cdots\cup[x_{n-2},x_{n-1}]\cup[x_{n-1},x_n]](https://tex.z-dn.net/?f=%5Bx_0%2Cx_1%5D%5Ccup%5Bx_1%2Cx_2%5D%5Ccup%5Ccdots%5Ccup%5Bx_%7Bn-2%7D%2Cx_%7Bn-1%7D%5D%5Ccup%5Bx_%7Bn-1%7D%2Cx_n%5D)
where
and 
. Each subinterval has measure (width) 
.
Now denote the left- and right-endpoint approximations by
and 
, respectively. The left-endpoint approximation consists of rectangles whose heights are determined by the left-endpoints of each subinterval. These are 
. Meanwhile, the right-endpoint approximation involves rectangles with heights determined by the right endpoints, 
.
So, you have
Now let
denote the trapezoidal approximation. The area of each trapezoidal subdivision is given by the product of each subinterval's width and the average of the heights given by the endpoints of each subinterval. That is,
Factoring out
and regrouping the terms, you have
which is equivalent to
and is the average of and .
So the trapezoidal approximation for your problem should be
Let's consider a simple example of estimating the value of a general definite integral,
Split up the interval
where
Now denote the left- and right-endpoint approximations by
So, you have
Now let
Factoring out
which is equivalent to
and is the average of
So the trapezoidal approximation for your problem should be