(a) Determine a cubic polynomial with integer coefficients which has $\sqrt[3]{2} + \sqrt[3]{4}$ as a root.

(b) Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational.

Answers

Answer 1

Answer:

(a) [tex]x\³ - 6x - 6[/tex]

(b) Proved

Step-by-step explanation:

Given

[tex]r = $\sqrt[3]{2} + \sqrt[3]{4}$[/tex] --- the root

Solving (a): The polynomial

A cubic function is represented as:

[tex]f = (a + b)^3[/tex]

Expand

[tex]f = a^3 + 3a^2b + 3ab^2 + b^3[/tex]

Rewrite as:

[tex]f = a^3 + 3ab(a + b) + b^3[/tex]

The root is represented as:

[tex]r=a+b[/tex]

By comparison:

[tex]a = $\sqrt[3]{2}[/tex]

[tex]b = \sqrt[3]{4}$[/tex]

So, we have:

[tex]f = ($\sqrt[3]{2})^3 + 3*$\sqrt[3]{2}*\sqrt[3]{4}$*($\sqrt[3]{2} + \sqrt[3]{4}$) + (\sqrt[3]{4}$)^3[/tex]

Expand

[tex]f = 2 + 3*$\sqrt[3]{2*4}*($\sqrt[3]{2} + \sqrt[3]{4}$) + 4[/tex]

[tex]f = 2 + 3*$\sqrt[3]{8}*($\sqrt[3]{2} + \sqrt[3]{4}$) + 4[/tex]

[tex]f = 2 + 3*2*($\sqrt[3]{2} + \sqrt[3]{4}$) + 4[/tex]

[tex]f = 2 + 6($\sqrt[3]{2} + \sqrt[3]{4}$) + 4[/tex]

Evaluate like terms

[tex]f = 6 + 6($\sqrt[3]{2} + \sqrt[3]{4}$)[/tex]

Recall that: [tex]r = $\sqrt[3]{2} + \sqrt[3]{4}$[/tex]

So, we have:

[tex]f = 6 + 6r[/tex]

Equate to 0

[tex]f - 6 - 6r = 0[/tex]

Rewrite as:

[tex]f - 6r - 6 = 0[/tex]

Express as a cubic function

[tex]x^3 - 6x - 6 = 0[/tex]

Hence, the cubic polynomial is:

[tex]f(x) = x^3 - 6x - 6[/tex]

Solving (b): Prove that r is irrational

The constant term of [tex]x^3 - 6x - 6 = 0[/tex] is -6

The divisors of -6 are: -6,-3,-2,-1,1,2,3,6

Calculate f(x) for each of the above values to calculate the remainder when f(x) is divided by any of the above values

[tex]f(-6) = (-6)^3 - 6*-6 - 6 = -186[/tex]

[tex]f(-3) = (-3)^3 - 6*-3 - 6 = -15[/tex]

[tex]f(-2) = (-2)^3 - 6*-2 - 6 = -2[/tex]

[tex]f(-1) = (-1)^3 - 6*-1 - 6 = -1[/tex]

[tex]f(1) = (1)^3 - 6*1 - 6 = -11[/tex]

[tex]f(2) = (2)^3 - 6*2 - 6 = -10[/tex]

[tex]f(3) = (3)^3 - 6*3 - 6 = 3[/tex]

[tex]f(6) = (6)^3 - 6*6 - 6 = 174[/tex]

For r to be rational;

The divisors of -6 must divide f(x) without remainder

i.e. Any of the above values  must equal 0

Since none equals 0, then r is irrational


Related Questions

which inequality is represented on the number line shown?

Answers

Answer: A x> -2

Step-by-step explanation:

It took francisco 60 minutes to walk from his house to his grandmother’s house. what is 60 written as a product of factors greater than 1? each factor can have only 1 and itself as factors.

Answers

Answer:

2 × 2 × 3 × 5

Step-by-step explanation:

Given that,

The number = 60

To find,

Factors of 60 greater than 1 = ?

Procedure:

As we know,

Any of various numbers multiplied together to form a whole.

To find the factors of a number, we will have to do its prime factorization.

So,

The prime factorization of 60:

1 * 2 * 2 * 3 * 5 = 60

Since the factors greater than 1 are asked, the factors would be;

2 * 2 * 3 * 5

Thus, 2 * 2 * 3 * 5 is the correct answer.

can i get some help solving this

Answers

Answer:

A =147 cm^2

Step-by-step explanation:

A = pi r^2

The radius is 7 and let pi = 3

A = 3*7^2

A = 3*49

A =147

A. -5x+4y=-20
B. -5x-4y=-20
C. -5x+4y=0
D. 5x+4y=-20

Answers

A is the answer because, when graphing, we use y= mx + b
So the first step is to change the equation into that.
We know m = the slope and b = the y-intercept

So,
m = y1-y2/x1-x2 ( I decided to use the intercepts)

The rest is on the paper, and sorry I wasn't neat with my working

Rihanna works in a coffee shop approximately nine miles from her apartment She bikes every day from her apartment to the coffee shop and then back to her apartment in the evening. However, on her way to work, she always stops halfway through to meet her best friend at a park. If the distance from apartment to the park is (5x +2) miles and the trip from the park to the coffee shop is (25x -8) miles long, then what is the value of x?

Answers

Answer:

x = 0.5

Step-by-step explanation:

When we add up the distance between the apartment and park with the distance between the park and the coffee shop, the total will equal nine miles.

Our equation will look like this:

(5x + 2) + (25x - 8) = 9

Add up the like terms

5x + 2 + 25x - 8 = 9

30x - 6 = 9

Add 6 to both sides

30x - 6 = 9

      + 6  + 6

30x = 15

Divide both sides by the coefficient, 30, to isolate the variable, x

30x/30 = 15/30

x = 15/30

Reduce

x = 1/2

A garden is rectangular with a width of 8 feet and a length of 10 feet. If it is surrounded by a walkway 2 feet wide, how many square feet of area does the walkway cover?

Answers

Answer:

The rea of walk way is 32 ft^2.

Step-by-step explanation:

width, w = 8 feet

length, L = 10 feet

width of walkway, d = 2 feet

length of outer, L' = 10 + 2 + 2 = 14 feet

Area of outer, A' = L' x w = 14 x 8 = 112 ft^2

Area of inner, A = L x w = 10 x 8 = 80 ft^2

The area of walkway = A' - A = 112 - 80 = 32 ft^2

The rea of walk way is 32 ft^2.

Helpo pleasssse
On my hw I have a parabola that opens down with its vertex at (-3,-6)......
For the range would I say that {yER | y > -6} OR {yER | y < -6} ????
I'm just confused from the negative numbers

Answers

Answer: The range is  [tex]\{y \in \mathbb{R}\ | \ y \le -6\}[/tex]

Explanation:

The parabola opens down, forming a "frowny face" in a way (just without the eyes). Or you can think of it as a hill or mountain. This means that the vertex (-3,-6) is at the top of that mountain. It's the highest point of that parabola.

The range is the set of all possible y values. We see that y = -6 is the largest it can get. So y = -6 or y is smaller than this. We would then write [tex]y \le -6[/tex] to describe all the possible y values.

Therefore, the range is [tex]\{y \in \mathbb{R}\ | \ y \le -6\}[/tex]

This translates to "y is a real number such that y is -6 or smaller".

So the second answer you wrote is close, but you forgot the "or equal to" portion of the inequality sign.

See below for a visual example of what's going on.

Match function with its corresponding graph

Answers

Answer:

Step-by-step explanation:

We can see that there are roots at (-2,0) and (-1,0)

also, the root at (-2,0) should bounce right off

and the root at (-1,0) should go through

With all that being said it has to be B

y=8200(0.96)^x growth or decay find

Answers

Answer:

This would be a .04 or 4% decay.....

for every "time unit" (x in this case) you will be multiplying

the amount by .96 ... in other words if you started with one dollar

the results would be 96 cents... after two "time" steps you would have

only 92 cents (.96 *.96)

Step-by-step explanation:

Can someone help me on this please

Answers

I’m pretty sure it would be K because it is a constant with units of dollars per square foot.
Hope this helps :D

Please someone tell me the answer of these questions

Answers

Answer:

VERTICALLY OPP ANGLES

Step-by-step explanation:

32
Two forces one is 10N and other is 6N act on a body The directions are unknown the resultant force on the body is
a. between 4 and 16N
b. more than 6N I
c. more than 1ON
d between 6 and 16N

Answers

If Forces are acting on opposite direction

[tex]\\ \rm\hookrightarrow F_{net}=F_2-F_1[/tex]

[tex]\\ \rm\hookrightarrow F_{net}=10-6[/tex]

[tex]\\ \rm\hookrightarrow F_{net}=4N[/tex]

If both acting on same direction

[tex]\\ \rm\hookrightarrow F_{net}=F_1+F_2[/tex]

[tex]\\ \rm\hookrightarrow F_{net}=10+6[/tex]

[tex]\\ \rm\hookrightarrow F_{net}=16N[/tex]

Hence

[tex]\boxed{\sf 4N\leqslant F_{net}\leqslant 16N}[/tex]

pls pls pls pls help

Answers

Step-by-step explanation:

[tex]s = \pi \times {r}^{2} = \pi \times {6}^{2} = 36\pi[/tex]

[tex]h = 18 \times \sin(60) = 9 \sqrt{3} [/tex]

[tex]v = s \times h = 36\pi \times 9 \sqrt{3} = 324 \sqrt{3} \pi[/tex]

-1/2 divided by 1/19

Answers

Answer:

-19/2 or

-9 1/2

Step-by-step explanation:

-1/2 ÷ 1/19

Copy dot flip

-1/2 * 19/1

-19/2

-9 1/2

(06.01)A scatter plot is shown:

A scatter plot is shown. Data points are located at 1 and 8, 2 and 7.8, 3 and 7.4, 4 and 6.5, 5 and 5.4, 6 and 4.5, 7 and 3.1, 8 and 2, 9 and 1.

ASAP:
What type of association does the graph show between x and y?

Linear positive association
Nonlinear positive association
Linear negative association
Nonlinear negative association

Answers

Answer:

Step-by-step explanation:

If you plot those coordinates in your calculator in the stat plot function under "stat", you will see that the dots are almost but not quite in a straight line going from upper left to lower right. This indicates a strong negative linear association, third choice down.

Answer:

D. Nonlinear Negative Association

Step-by-step explanation:

Suppose that d varies jointly with r and t, and d = 110 when r = 55 and t = 2. Find r when d = 40 and t = 3.

Answers

Answer:

r = 13.33

Step-by-step explanation:

d = k*r*t

Where,

k = constant of proportionality

d = 110 when r = 55 and t = 2

d = k*r*t

110 = k * 55 * 2

110 = 110k

k = 110/110

k = 1

Find r when d = 40 and t = 3

d = k*r*t

40 = 1 * r * 3

40 = 3r

r = 40/3

= 13.333333333333

Approximately,

r = 13.33

A giant pie is created in an attempt to break a world record for baking. The pie is shown below:

What is the area of the slice of pie that was cut, rounded to the nearest hundredth?

Answers

Answer:

Area of the slice of pie = 22.09 ft²

Step-by-step explanation:

Area of the slice of pie = Area of the sector of the circle with the central angle 45°

Area of the sector = [tex]\frac{\theta}{360^{\circ}}(\pi r^{2} )[/tex] [Here, r = radius of the circle]

                              = [tex]\frac{45^{\circ}}{360^{\circ}}(\pi )(\frac{15}{2})^2[/tex]

                              = 22.09 ft²

Area of the slice of pie = 22.09 ft²

Answer:

22.08ft^2

Step-by-step explanation:

A = πr^2(x/360) d = 15

Since r is half of diameter this means that r = 15/2 =7.5

so Lets use the Area of Sector formua

A =3.14(7.5)^2 (45/360)

A =3.14(56.25) (45/360)

A = 176.625 (45/360)

A = 176.625 (0.125)

A = 22.078125

rounded to the nearest 10th would make it 22.08

b) 2x (x - y) + 3y (x - y)​

Answers

Use distributive law

[tex]\boxed{\sf a(b+c)=ab+ac}[/tex]

Now

[tex]\\ \sf\longmapsto 2x(x-y)+3y(x-y)[/tex]

[tex]\\ \sf\longmapsto 2x^2-2xy+3xy-3y^2[/tex]

[tex]\\ \sf\longmapsto 2x^2-3y^2-2xy+3x^2[/tex]

[tex]\\ \sf\longmapsto 2x^2-3y^2+xy[/tex]

Taking common

Answer: 2x (x-y) + 3y (x-y)

             = ( x-y ) ( 2x-3y )

Grace bought a property valued at $200,00.00 and 20% down and a mortgage amortized over 10 years. She makes equal payments due at the end of every months. Interest on the mortgage is 4% compounded semi-annually and the mortgage is renewable after five years. a) What is the size of each monthly payment? b) What is the outstanding principal at the end of the five-year term? c) What is the cost of the mortgage for the first five years?

Answers

Answer:

Size of each monthly payment = $161.69 per month

Step-by-step explanation:

Given:

Value of property = $20,000

Downpayment = 20%

Number of payment = 12 x 10 = 120

Interest rate = 4% = 4% / 12 = 0.33 %

Computation:

Loan balance = 20,000 - 20%

Loan balance = $16,000

A] Size of each monthly payment [In Excel]

Size of each monthly payment = PMT(0.33%,120,16000,0)

Size of each monthly payment = $161.69 per month

(b) A shopkeeper gives 20% discount on the marked price of a television set. The VAT amount at the rate of 13% is Rs. 2,600. Find the marked price and the amount of discount.​

Answers

Answer:

6.3

Step-by-step explanation:

i had the same question for school

A walker has travelled 9 km along a trail. If he has completed 80% of the trail, how much further does he still have to go?

Answers

Answer:

2.25 km to go

Step-by-step explanation:

In order to get this answer, you have to figure out how many km 10% is, 0.1125. Then multiply that by the remaining 20% because he already finished 80% of the trail. So, 0.1125 x 20 = 2.25.

Hope this helps! :)

The perimeter of a rectangle is 18cm . if the length is (x+2), find it's width.​

Answers

Answer:

W = 7 - x

Step-by-step explanation:

The perimeter is P= 2L + 2×W , where L is the length and W is the width.

If L = (x+2) , replacing L with the expression x+2 we have

P= 2×(X+2) + 2W ⇔ 18 = 2x + 4 + 2W ⇔ 2W =18 - 2x - 4 ⇔ 2W = 14 - 2x

⇔ W = 7 - x

At a particular restaurant, each slider has 225 calories and each chicken wing has 70 calories. A combination meal with sliders and chicken wings has a total of 7 sliders and chicken wings altogether and contains 1110 calories. Write a system of equations that could be used to determine the number of sliders in the combination meal and the number of chicken wings in the combination meal. Define the variables that you use to write the system.

Answers

Answer:

X+y=7

Step-by-step explanation:

i  remember  doing something like this but mines had the word onion rings .

What's 672 divided by 32

Answers

the answer is 21, blessup my friend stay groovy

please ans this question pleaseee​

Answers

Answer:

[tex]{ \tt{ \tan {}^{4} \theta + { \sec }^{2} \theta }} \\ { \tt{ = ({ \tan }^{2} \theta ){}^{2} + { \sec }^{2} \theta }} \\ = { \tt{ {-(1 - { \sec }^{2} \theta) }^{2} + { \sec }^{2} \theta }} \\ { \tt{ = -(1 - 2 { \sec }^{2} \theta + { \sec }^{4} \theta) + { \sec}^{2} \theta}} \\ { \tt{ = -(1 - { \sec }^{2} \theta) + { \sec }^{4} \theta}} \\ { \tt{ = -{ \tan}^{2} \theta + { \sec }^{4} \theta }} \\ = { \tt{ { \sec}^{4} \theta - { \tan }^{2} \theta}} \\ { \bf{hence \: proved}}[/tex]

The point B (-2, 1) has been transformed to B' (-5, -3). The transformation is described as
A T (-3,-4)
B R x=2
C T (-3,-2)
D D3

Answers

Answer:

Option A: T(-3, -4)

Step-by-step explanation:

For a general point (x, y), if we apply the transformation:

T(a, b)

at that point, the new point that we will get is:

T(a,b)[ (x, y) ] = (x + a, y + b)

Notice that if w take the difference between the new point (x + a, y + b) and the original point (x, y)

we get:

(x + a, y + b) - (x, y) = (x + a - x, y + b  y) = (a, b)

These are x-value and y-value that describe our transformation T(a, b).

Then if we know that the original point is B (-2, 1)

and the transformed point is B'( -5, -3)

We can just take the difference to get:

(-5, -3) - (-2, 1) = (-5 - (-2), -3 - 1) = (-5 + 2, -4) = (-3, -4)

Then the transformation applied is:

T(-3, -4)

The correct option is A.

For this exponential function,
what is the output value (y),
when the input value (x) is 3?
y = 10.5x.
(3, [?])

Answers

Replace x with 3 and solve:

10 x 5^3

Simplify:

10 x 125 = 1250

Answer: y = 1250

Answer:

(3 , 1250)

Step-by-step explanation:

I got it right on my test.

Find all solutions to the equation

in the interval [0, 21). Enter the

solutions in increasing order.

Answers

Answer:

0,1,2,3,4,5,6.......19,20

Step-by-step explanation:

in this interval 0 is included and 21 is not included. So starting from 0 up to 20, all are the solutions

9. Find the remainder when the polynomial: p(x) = x⁴ + 2x³- 3x² + x - 1 is divided by (x - 2)
pls it's urgent

Answers

Answer:

answer is 21..............

Explanation:

p(x) = x⁴ + 2x³- 3x² + x - 1

Factor of p(x)

x-2=0

x=2

Then by using synthetic division

sin x = 4/5, cos x = 2/5 find the value of tan x​

Answers

Answer:

2 is the answer . the explanation is in the attachment .

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