Prove that if f(x) ε F[x] is not irreducible, then F[x] / contains zero-divisors.

Answers

Answer 1

if f(x) ε F[x] is not irreducible, then F[x]/ contains zero-divisors.

Suppose that f(x) is not irreducible in F[x]. Then we can write f(x) as the product of two non-constant polynomials g(x) and h(x), where the degree of g(x) is less than the degree of f(x) and the degree of h(x) is less than the degree of f(x).

Therefore, in F[x]/(f(x)), we have:

g(x)h(x) ≡ 0 (mod f(x))

This means that g(x)h(x) is a multiple of f(x) in F[x]. In other words, there exists a polynomial q(x) in F[x] such that:

g(x)h(x) = q(x)f(x)

Now, let us consider the images of g(x) and h(x) in F[x]/(f(x)). Let [g(x)] and [h(x)] be the respective images of g(x) and h(x) in F[x]/(f(x)). Then we have:

[g(x)][h(x)] = [g(x)h(x)] = [q(x)f(x)] = [0]

Since [g(x)] and [h(x)] are non-zero elements of F[x]/(f(x)) (since g(x) and h(x) are non-constant polynomials and hence non-zero in F[x]/(f(x))), we have found two non-zero elements ([g(x)] and [h(x)]) in F[x]/(f(x)) whose product is zero. This means that F[x]/(f(x)) contains zero-divisors.

To learn more about polynomials visit:

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Related Questions

You park in a garage 3 floors below ground level. Then you get in the elevator and go up 12 floors. What floor do you get off the elevator?

Answers

Answer: (B) 6 floors

Step-by-step explanation: Use the equation -3 + 9 to get the amount of floors above ground level.

-3 + 9 = 6

6 floors.

If this helps may I please have a brainliest.

Answer:

you would get off at the 9th floor because in order to reach ground level again your would already go up 3 of the 12 floors. leaving 9 floors left to go up.

Step-by-step explanation:

-3+12=9

A company has 200 employees there are 150 female employees. 40 employees are under the age of 20. The probability of randomly selecting an employee who is female or is under the age of 30 is 85%. What is the probability of randomly selecting an employee who is female and under the age of 30?

Answers

Answer and explanation:

Probability= number of favorable outcomes/ total number of outcomes

Probability of selecting an employee who is female= number of female employees/ total number of employees

Probability of selecting an employee who is female= 150/200

Probability of selecting an employee who is under 30= number of employees under 20/ total number of employees

Probability of selecting an employee who is under 30 = 40/200

Probability of randomly selecting an employee who is female and under the age of 30= Probability of selecting an employee who is female × Probability of selecting an employee who is under 30

Probability of randomly selecting an employee who is female and under the age of 30= 150/200×40/200=15/20×4/20=3/20= 0.15

Therefore there's a 15% chance of selecting an employee who is female and under the age of 30

The right triangle below has legs of length and.
The hypotenuse has length.
Four copies of this triangle are arranged as follows.
The hypotenuses form a square of side length and area.
Answer the questions below to find how, and are related.
Part 1: Compute the total combined area of the four triangles:

Part 2: Compute the area of the large (outer) square:
Part 3: Using your answers in Parts 1 and 2, find the area of the small (inner) square.
Part 4: We are given the side lengths and. Compute.
Part 5: Use, or complete the statement below.

Answers

Answer + explanation:

Part 1: First, find the area of one triangle using the formula base * height / 2. You have base = 10 and height = 7, so area = 7 * 10 /2 = 70/2 = 35. Since you're finding the combined area of 4, you have to do 35 * 4 = 140.

Part 2: To do this, find the side length of the square. Looking at the picture, you see that all you have to do is 10 + 7 = 17 to find one side. Then, to find the area do 17 * 17 = 289.

Part 3: To find the area of the small square, all you do is area of the large square - combined area of 4 small triangles, so 289 - 140 = 149.

Part 4: a^2 = 7 * 7 = 49. Then, b^2 = 10 * 10 = 100. Adding them together, you get 100 + 49 = 149.

Part 5: Since you already have the a^2 + b^2 part, all you need to do is find what c^2 is. You can see in the picture that c^2 equals the area of the small square, which you found in part 3. c^2 = 149. Then, you have 149 and 149, which are equal so the answer is =.

Daniel wants to know the volume of his cube. He notices that the cube 1/5 centimeters side lengths. What is the volume of the cube?

Answers

Answer:

1/125 cm^3

Step-by-step explanation:

Given data

Length of one side = 1/5cm

Required

The volume of the cube

We know that

V= l^3

substitute

V= (1/5)^3

V= 1/125cm^3

Hence the volume is 1/125 cm^3

The residents of a village want to paint a sign that shows the elevation of their village to the sea. Their altimeter shows +2,100 feet. What should their signboard read?

Answers

Answer:

Wassss uppppp

Step-by-step explanation:

i dont know

I NEED HELP WITH THESE ASAP!

Answers

Answer:

I can not see it

Step-by-step explanation:

Can some one solve this

Answers

Answer:

J Monica is right

Step-by-step explanation:

Add three to both sides, divide by 4 you get x[tex]\geq[/tex]4

Answer:

J

Step-by-step explanation:

3/11 as a decimal equivalent

Answers

0.27 repeated with the line over the 27

Answer:

The decimal form for this is 0.27 (With a line on top of it.)

Step-by-step explanation:

See this image:

Figs come in 2.4 lb bags. Apples come in 7.2 lb bags. How many times as many pounds are the apple bags as the fig bags?

Answers

Answer:

3 times as many pounds

Step-by-step explanation:

7.2 / 2.4 = 3 times as many pounds

Hope that helps!

Which equations are correct?

Select each correct answer.




−4b3(5b2+3)=−20b6−12b3

−6y4(4y2+2)=−24y8−12y4

−5a4(2a2+4)=−10a6−20a4

−4x2(2x2+5)=−8x4−20x2

Answers

Option C

-5a⁴(2a² + 4)

-10a⁶ - 20 a⁴

Find the length of AD if AB = 24, DE = 8, and DC = 12.
a.8
b.12
c.14
d.10

HELP PLS

Answers

Answer:

AD = 12

Step-by-step explanation:

Find the diagram attached

From the given diagram, it can be inferred that;

AB = AD+DB and AD = DB

AB = AD + AD

AB = 2AD

AD = AB/2

Given that AB = 24

AD = 24/2

AD = 12

Hence the length of AD is 12

Answer: 10

Step-by-step explanation:

I got it right after trying the answer of 12 on CAOLA

What’s the volume of this cube?

Answers

Answer:

Formula: V = a^3

a = 2in

Input: V = a^3 = 2^3 = 8

Step-by-step explanation:

A baseball player earns $3 million in 2015. During the next 15 years, the earnings increase 4% each year.

a. Write an exponential growth model giving the earnings y (in millions of dollars) t years after 2015. Write the base of the model as a decimal.

An exponential growth model is y=
b. In which year will the player's earnings first exceed $5,000,000?

Answers

Answer:

[tex]y=3*1.04^{t}[/tex]

in 2029

Step-by-step explanation:

[tex]y=3*1.04^{t}[/tex]

[tex]when y=5\\t=\frac{ln(5/3)}{ln(1.04)} =13.02 ..[/tex]

have to round it up to 14 to exceed

which equation has a solution x= -3
A) 2x - 7 = -1
B) 3x + 8 = 1
C) 1/2x + 8 = 10
D) 1/2(2x - 6) = -6

Answers

Answer:

D

Step-by-step explanation:

1/2 ( 2x - 6) = -6

1/2 = 0.5

0.5(2x-6) = -6

1x-3=-6

1x=-6+3

x=-3/1

x=-3

Answer is D

Step by step:

1/2(2x - 6) = -6
-6 -6
1/2 (-12) = -6
-6 = -6

Transform to slope-intercept form. 1+2y=6x-3

Answers

Hello the slope intercept form is y=3x-2

Answer: Y=3X-2

Step-by-step explanation:

First you are gonna need to combine all like factors.

1+2y=6x-3 (subtract 1 from both sides)

2y=6x-4

Then you need to get the y by itself.

2y=6x-4 (divide all terms by 2)

y=3x-2

now you are in slope intercept form (y=mx+b)

help me please I need help

Answers

Answer:

D) 6 in x 4 in

Step-by-step explanation:

3" x 2" scaled up to 1:20 would be 60" x 40"

60" x 40" scaled back to 1:10 would be 6" x 4"

also the scale factor 1:20 is double the scale factor of 1:10

Pensé dos números. Sumados dan 38 y restados dan 2. ¿Qué números son?

Answers

Answer:

sorry I cannot help you sir can you translate it into English

Help Please........ ​

Answers

Answer:

4

Step-by-step explanation:

Answer:

12

Step-by-step explanation:

(4 x 2)/2 + (4 x (6-2))/2 = 12

Solve for x.2(-x - 4) = 4x + 16
a) 4
b) 3
c) -3
d) -4

Answers

Answer:

44

Step-by-step explanation:

Answer:

X=-4

Step-by-step explanation:

First, we must solve for the parenthesis so...

-2x-8=4x+16

Then we must simplify this even more so lets add the 8 to 16

-2x=4x+24

Then we must subtract 4x from -2x

-6x=24

we must now divide 24 by -6 so we can isolate the x

x=-4

I hope this helped :)

Which polynomial represents the difference below?
2x +7X+6
(3x? – X
A. - x + 6x + 6
B. 2x2 + 4x + 6
c. 2x2 + 5x + 6
D. - x2 + 3x + 6

Answers

Answer:

-x²-8x+6

Step-by-step explanation:

Given the polynomial expression

2x²+7x+ 6 and 3x² – X

Taking their difference

2x²+7x+ 6  - (3x² – X)

2x²-3x²+7x+x+6

-x²-8x+6

Hence the required difference is -x²-8x+6

Answer:

-x²+8x+6

Step-by-step explanation:

Ape-x

can someone explain how to solve this please my midterms tomorrow! :)

Answers

Answer:

Step-by-step explanation:

Convert the decimal below to a fraction in simplest form. 0.689

Answers

Answer:

689 / 1000 is the answer

Step-by-step explanation:

zzzzzzzzzzzzz

Can somone help please? Thx​

Answers

Answer:

I would say the answer is 0.69

Step-by-step explanation:Because if you multiply 7 times 0.69 it gives you 4.83

Answer:

The answer is 1.45

Step-by-step explanation:

Divide 7 and 4.83 and the answer is 1.447.

help plz
−86 + (−14) − 23 − (+22) + 16 − (−34)=​

Answers

Step-by-step explanation:

the answer is -86 mark as brainliest or rate this if u found this helpful :)

Answer:

-105

Step-by-step explanation:

-86-14= -100

-100-55= -155

16+34= 50

-155+50= -105

what does the ! after a number mean.
explain in your own words and give 5 examples.

Answers

Answer: it means that you multiply the number by the numbers counting to 0

Step-by-step explanation:4!=4•3•2•1

-4x < 28
does anyone know the solution and answer?​

Answers

Answer:

x>−7

Explanation:

Let's solve your inequality step-by-step.

−4x<28

Step 1: Divide both sides by -4.

−4x/−4 < 28/−4

x>−7

Answer: x>−7

the base of a rectangular prism has an area of 1 over 8 ft2 and the height of the prism is 2 over 3 ft. what is the volume of the prism?​

Answers

Answer:  the volume is 1/12 ft³

Step-by-step explanation:

Multiply the area of the base times the height:

1/8 × 2/3 = 2/24 or 1/12

the ft² times ft becomes ft³

So the volume is 1/12 ft³

4w − 18 = 18 solve for w

Answers

Answer:

hello! :) have a good day!

W = 9

which one pls help me i’ll give extra points

Answers

Answer:

C

Step-by-step explanation:

Answer:

A. The doctors will give Charlie an operation

Step-by-step explanation:

Plz help me !!!!!!!!!!

Answers

C i think?????????????????????  :D

C is the answer ! :)
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