Which probabilities do you need in order to

calculate P(getting A on 2 or fewer spins)?

-P(0 successes)

-P(1 success)

-P(2 successes)

-P(3 successes)

-P(4 successes)

-P(5 successes)

Answer: A,B, C


part B:

P(getting A on 2 or fewer spins)= Answer: 0. 79

The final result of the subtraction is: -4 ÷ (1 - 5[tex]x^{2}[/tex])

What is Algebraic expression ?

A cοmbinatiοn οf variables and cοnstants is an algebraic expressiοn.

To subtract the expression:

(-10 ÷ (2 - 10[tex]x^{2}[/tex])) - (-2 ÷ (2 - 10[tex]x^{2}[/tex])))

we need to first simplify the denominator by factoring out a common factor of 2:

2 - 10[tex]x^{2}[/tex]= 2(1 - 5[tex]x^{2}[/tex])

Now we can write the expression as:

(-10 ÷ [2(1 - 5[tex]x^{2}[/tex])]) - (-2 ÷ [2(1 - 5[tex]x^{2}[/tex])])

which simplifies to:

(-5 ÷ [1 - 5[tex]x^{2}[/tex]]) - (-1 ÷ [1 - 5[tex]x^{2}[/tex]])

Using the fact that subtracting a negative is the same as adding a positive, we can rewrite this as:

(-5 + 1) ÷ [1 - 5[tex]x^{2}[/tex]]

which equals:

-4 ÷ [1 - 5[tex]x^{2}[/tex]]

Therefore, the final result of the subtraction is: -4 ÷(1 - 5[tex]x^{2}[/tex])

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Complete Answer:

Subtract the expression [tex]$(-10\div(2-10x^{2}))-(-2\div(2-10x^{2}))$[/tex]

a) July 2- $5600, August 2- $5200, September 2- $4800, October 2- $4400, November 2- $4000

b) The amount owed each month forms an arithmetic sequence because it decreases by the same amount ($400) each month.

For the month of July, the amount owed is

6,000 - 400(1-1) = 6,000 - 400(0) = 6,000 - 0 = $6,000.

For the month of August, the amount owed is

6,000 - 400(2-1) = 6,000 - 400(1) = 6,000 - 400 = $5,600.

For the month of September, the amount owed is

6,000 - 400(3-1) = 6,000 - 400(2) = 6,000 - 800 = $5,200.

For the month of October, the amount owed is

6,000 - 400(4-1) = 6,000 - 400(3) = 6,000 - 1,200 = $4,800.

For the month of November, the amount owed is

6,000 - 400(5-1) = 6,000 - 400(4) = 6,000 - 1,600 = $4,400.

The amount owed by Mr. Taylor on the 2nd of each month starting with July 2 is as follows: July 2- $5600, August 2- $5200, September 2- $4800, October 2- $4400, November 2- $4000. This forms an arithmetic sequence because each month the amount owed decreases by the same amount of $400.Arithmetic sequences are collections of integers where each term following the first is created by adding a predetermined constant to the term before it. In this case, the constant is $400 because each month the amount owed is decreased by $400. An arithmetic sequence can be written as a mathematical expression, with the nth term being expressed as an + d(n-1). In the case of Mr. Taylor, the initial amount (a) is $6,000 and the common difference (d) is -$400 because the amount owed decreases each month. Therefore, each month the amount owed is expressed as 6,000 - 400(n-1).

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If the ratio of cows to sheep is 3:7, then there are 210 sheep on the farm if there are 90 cows.

The first step is to write out the parameters:

Now, the number of sheep on the farm can be calculated as follows:

Hence there are 210 sheep on the farm

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Simulation shows Luke's quiz score in 50 trials with a minimum passing score of 6. Probability of passing is about 0.1. Strong evidence that Luke understands some material as getting a score of 8 by guessing is highly unlikely.

Step 1: The teacher plans to give a multiple-choice test consisting of 10 questions with four answer options each. To conduct one trial of the simulation, a random number generator will be used. To pass the test, the student needs to answer at least six questions correctly. Since each question has four options, the probability of guessing the correct answer is one out of four. To simulate guessing, ten numbers will be generated between one and four, where one represents a correct answer and 2, 3, 4 represent incorrect answers.

Step 2: The dot plot provided illustrates the simulated quiz score of Luke through 50 trials. Each dot on the plot corresponds to a single trial. One dot, specifically the one located at 1, represents a simulated quiz score of one. This implies that in one of the simulation trials, Luke answered only one out of ten questions correctly.

Step 3: We must determine the likelihood of Luke passing the quiz, which contains 10 multiple-choice questions with four answer choices each, requiring a minimum of 6 correct answers to pass. In the dot plot depicting 50 trials of the simulation, the dots represent the quiz score, and 5 of the 50 trials resulted in a score of 6 or more. Thus, the probability of Luke passing the quiz is approximately 0.1 or 1/10.

Step 4: We need to determine whether there is sufficient evidence to support the claim that Doug has an understanding of some of the material.

Step 5: The dot plot displays the simulated quiz score over 50 trials when a person is guessing the answers to the questions. We can infer that it is highly improbable to obtain a quiz score of at least 8 by randomly guessing, as there are no dots above or to the right of 8 on the plot. This suggests that there is compelling evidence that the person understands some of the subject matter, as it is unlikely that they guessed correctly on all questions.

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The missing figure is in the image attached below

The angle that has the sine ratio, 4/5, is: A. Angle P.

Sine Ratio:

The sine ratio is the ratio of the side opposite the hypotenuse of a right triangle to a given reference angle. When the ratio is found using the opposite and the hypotenuse, we call it the sine ratio rather than the tangent ratio.

Example:

Then we say that the sine of 45 degrees is equal to 0.707. In short, we can use the notation sin instead of sine and write sin(45 degrees) = 0.707

In general, the sine of angle A = leg length of opposite angle A / hypocenter

sin(A ) = opposite / hypotenuse

Given:

sine ratio of 4/5, then it means:

Opposite side (side directly opposite to the reference angle) = 4

Hypotenuse (longest side) = 5.

Thus, in the image given, using P as the reference angle, the sine ratio of P is:

QR/PQ

= 16/20 = 4/5

Therefore, the angle with the sine ratio of 4/5 is: A (P)

Complete Question:

Using the side lengths of △PQR and △STU, which angle has a sine ratio of 4/5?

A. P

B. Q

C. T

D. U

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The acceptable temperatures the building can reach can be represented by the following inequality:

75°F ≤ temperature ≤ 92°F

This inequality states that the temperature must be greater than or equal to 75°F, but less than or equal to 92°F. Any temperature within this range is acceptable.

To graph this inequality, we can use a number line with 75 and 92 marked as endpoints, and shade the region in between the two endpoints, including the endpoints themselves. This shaded region represents all the acceptable temperatures the building can reach, as shown below:

|-------|-------|-------|-------|-------|------> Temperature (°F)

                70      75      80      85      90      95

                                 <------ Shaded Region

In this graph, the shaded region between 75 and 92 represents all the temperatures that are acceptable for the building. Any temperature outside of this region is not acceptable and would cause the building to be too hot for staff to work in.

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The number of hazardous waste sites in State Y in the year 2000 was 21.

Let n be the number of hazardous waste sites in State Y.

Based on the problem statement, we have

2n - 8 = 34

So, we have

2n = 42

Divide

n = 21

Hence, the waste sites in State Y in the year 2000 was 21.

Given that

6x + 1.6 = 58

Apply the subtraction property

6x = 56.4

Apply the division property

x = 9.4

Here, we have the bar diagram

Adding the terms in the bar, we have

n + n + n + 8 = 53

3n + 8 = 53

Apply the subtraction property

3n = 45

Apply the division property

n = 15

Hence, the bar equation is 3n + 8 = 53

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A cylinder has volume of 45 and radius 3, the height of the cylinder is 5 units.

The formula for the volume of a cylinder is given by V =

[tex]\pi r^2 h,[/tex]

where V is the volume, r is the radius, and h is the height.

In this case, we know that the volume is 45

[tex]\pi[/tex]

and the radius is 3. We can plug these values into the formula and solve for the height:

[tex]45 \pi = \pi (3)^2 h[/tex]

Simplifying the right-hand side of the equation:

Dividing both sides by 9

[tex]h = \frac{45 \pi}{9 \pi} = 5[/tex]

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So the convincing evidence that the radio station plays less than

[tex]50\ min\ of\ music\ per\ hour[/tex]. Here we have to see graph and chart.

Parameter of Interest, [tex]\mu = the\ true\ average\ number \ minutes\ of\ music \ played\ every \ hour.[/tex]

Null Hypothesis,  [tex]H_{o} : \mu = 50[/tex]

Alternative Hypothesis, [tex]H_{a} : \mu < 50[/tex]

[tex]Conditions\ of\ test :[/tex]

[tex]Random :[/tex] A random sample of [tex]hours[/tex] was selected.

[tex]Independent:[/tex] There are more than [tex]10(12) = 120\ hours[/tex] of music played during the week.

[tex]Normal:[/tex] We do not know if the population distribution of the music  [tex]times[/tex] is approximately Normal and we don’t have a large (big) sample size, so we will graph the data and look for any departures from Normality.

Level of Significance,  [tex]\alpha = 0.05\ Significance\ level[/tex]

[tex]n = 12, df = 11, \bar x = 47.9, S_{x} = 2.81[/tex]

1- var Stats:

[tex]\bar x= 47.9166[/tex] , [tex]\Sigma\ x = 575[/tex], [tex]\Sigma\ x^{2} = 27639[/tex], [tex]Sx = 2.81096[/tex], [tex]\sigma x = 2.69129[/tex]

[tex]t = \frac{\bar x- \mu_{o}}{ \frac{S_{x} }{\sqrt{n} } }[/tex]

[tex]t = \frac{47.9- 50}{ \frac{2.81 }{\sqrt{12} } }[/tex]

[tex]= - 2.59[/tex]

T test :

[tex]\mu < 50, t = -2.5674, p=.013, \bar x = 47.9166, S_{x} = 2.81, n = 12[/tex]

P-value (Use correct probability notation.) [tex]P-value = P(t < -2.59) = 0.0126[/tex]

Since the [tex]P-value(.013)[/tex] is less than [tex]\alpha =.05[/tex], we reject the null hypothesis.

There is convincing evidence(proof) that the radio station plays less than [tex]50\ min\ of\ music\ per\ hour[/tex].

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Option B is the true statement based on the given information by solving the method of average.

In mathematics, the average (also called the arithmetic mean) is a measure of central tendency that represents the sum of a set of numbers divided by the total number of values in the set.

Since both tests had a median score of 78, we know that there were nine scores below 78 and nine scores above 78 on each test.

If the original test had a range of 20, that means the highest score was 20 points above the lowest score. Therefore, the lowest score on the original test was 78 - 10 = 68, and the highest score was 68 + 20 = 88.

If the retest had a range of 2, that means the highest score was only 1 point above the lowest score. Therefore, the lowest score on the retest was 78 - 1 = 77, and the highest score was 77 + 2 = 79.

We don't know the mean score for either test, so we cannot determine if option A is true or false. We also don't know if any student scored exactly 78 on the retest, so we cannot determine if option C is true or false. Finally, we know that the highest score on the retest is 79, so option D is false.

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The answer fοr the imaginary expressiοn is 4i (οptiοn D)

Imaginary numbers are numbers that are nοt real. We knοw that the quadratic equatiοn is οf the fοrm ax² + bx + c= 0, where the discriminant is b²-4ac. In imaginary expressiοn οr number the discriminant becοmes negative οr less than 0.

Imaginary numbers are the numbers that give a negative number when squared. In οther wοrds, we can say that an imaginary number is basically the square rοοt οf a negative number which dοes nοt have a tangible value.

√-16

=√16i² [since i²= -1]

= 4i

Hence the value οf the imaginary number is 4i

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Answer: 4i and i√16

Step-by-step explanation: Just did it.

Answer:

Below

Step-by-step explanation:

2x + 3y =3       since  x = 6y+24  put that in for 'x'

2 ( 6y+24) + 3y = 3

12 y + 48 + 3y = 3

15 y + 48 = 3

15 y = -45

y = -3     <======use this value of 'y' in one of the equations to calculate the corresponding 'x' value :

 x = 6y + 24

 x = 6(-3) + 24

 x = 6

Answer: The quotient is 150 with a remainder of 34

Step-by-step explanation:

Answer:78.9

Step-by-step explanation:

78.9x 0896968

For example, if the estimated population proportion of adult office workers who have worn a Halloween costume to the office at least once is 0.25 and the sample size is 400, the margin of error would be calculated as follows:

[tex]ME = 1.96*sqrt((0.25*(1-0.25))/400) = 0.032.[/tex]

The margin of error is 0.032. This means that if you were to survey a sample of adult office workers about the proportion of them who have worn a Halloween costume to the office at least once, the results of the survey would be within plus or minus 0.032 of the true population proportion 95% of the time.

To calculate this margin of error,

use the following formula:[tex]ME = 1.96*sqrt((p*(1-p))/n)[/tex] where p is the estimated population proportion and n is the sample size.

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Answer:

3 hours

Step-by-step explanation:

to find the time the formula is

time=distance/speed

time=330km/110km/h

(the km will cancel out as it appears it the numerator and denominator)

time=330/110h

(330/110=3)

time=3hours

The absolute value inequality describing the scoring averages of the players is -6 <= x - 5 <= 6

The absolute value or modulus of a real number |0|=0}

If we remove the absolute value symbols from the following inequality

|expression| < value

then we get the following compound inequality

-value < expression < value

This is a basic property of absolute value inequalities that needs to be memorized. (If one understands absolute value to be a distance from zero on the Real number line, then this property is self-explanatory.)

Therefore, the inequality

|x - 5| <= 6

becomes the following compound inequality, once we use the property above to remove the absolute value symbols.

-6 <= x - 5 <= 6

Well, this process is reversible. I mean, once we come up with the numbers 5 and -6 and 6, and we write

-6 <= x - 5 <= 6

it's simply a matter of rewriting it using absolute value form

|x - 5| <= 6

So, now the question becomes: where did the -6, 6, and 5 come from?

Let x be a particular starter's scoring average.

Given: 4 <= x <= 10

The average of the endpoints of this given range of scoring averages is 15. I mean, the average of the lowest scoring average (4) and the highest scoring average (10).

Subtracting this average from each part in the given compound inequality above gets us to

-6 <= x - 5 <= 6

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Answer: S - The problem is asking us to find who made more money in 2022 with their Only Fans page and who has more subscribers. It also asks for recommendations for peers considering creating an Only Fans account.

O - The necessary facts to answer the problem are the number of paid monthly subscribers and the amount each subscriber pays for Aija and John.

L - To find out who made more money in 2022 with their Only Fans page, we can calculate the total income for Aija and John by multiplying the number of paid monthly subscribers by the amount each subscriber pays and then by the number of months. To find out who has more subscribers, we can compare the number of paid monthly subscribers for Aija and John.

V - Aija’s total income in 2022 = 560 subscribers * $14/subscriber * 12 months = $94,080 John’s total income in 2022 = $76,000 John’s number of subscribers = $76,000 / ($19/subscriber * 8 months) = 500 subscribers

E - Aija made more money in 2022 with her Only Fans page than John. Aija also has more subscribers than John. As for recommendations for peers considering creating an Only Fans account, it is important to carefully consider the potential risks and benefits before making a decision. It is also important to research and understand the platform’s terms of service and community guidelines.

(a) Trapezoid: Only one pair of parallel sides

(b) Rectangle: All sides congruent, Four right angles

(c) Parallelogram: Two pairs of parallel sides

(a) Trapezoid: Only one pair of parallel sides must be true. A trapezoid is defined as a quadrilateral with at least one pair of parallel sides, but the other two sides may or may not be congruent.

(b) Rectangle: All sides congruent and Four right angles must be true. A rectangle is a special case of a parallelogram where all angles are right angles and all four sides are congruent.

(c) Parallelogram: Two pairs of parallel sides must be true. A parallelogram is defined as a quadrilateral with two pairs of parallel sides. The opposite sides are congruent and parallel, but the adjacent sides may or may not be congruent.

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The estimated depth of the Grand Canyon would be approximately 467 feet.

First we need to calculate the height of the girl in inches. Since a foot is equal to 12 inches, the girl's height would be 5 x 12 = 60 inches. If the girl's shadow is 9 inches, then the ratio between the girl's height and her shadow is 60/9 or 6.6667 (rounded to 4 decimal places).Now, if the girl's shadow is 9 inches long, and she takes 280 steps to reach the end of it, and each step is approximately 3 feet long, then the total distance she has covered would be 280 x 3 = 840 feet.

The distance from the girl to the canyon is the height of the canyon. If we multiply the distance covered by the girl, which was 840 feet, by the ratio between the girl's height and her shadow length, which was 6.6667, we will get the height of the canyon. Therefore, the height of the Grand Canyon can be estimated to be 840 x 6.6667 = 5600 inches (rounded to the nearest whole number), which is equivalent to approximately 467 feet. Answer: The estimated depth of the Grand Canyon would be approximately 467 feet.

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The variance οf X is  2.716.

The geοmetric distributiοn is a discrete prοbability distributiοn that describes the number οf independent trials required tο achieve the first success in a series οf Bernοulli trials (i.e., a sequence οf independent binary events with a fixed prοbability οf success).

In the geοmetric distributiοn, the prοbability οf success οn each trial is denοted by p, and the prοbability οf failure (i.e., nοt achieving the desired οutcοme) is denοted by q = 1 - p. The randοm variable X represents the number οf trials required tο achieve the first success, and its probability distribution can be expressed as:

P(X = k) =  [tex]\mathrm{q^{(n-1)}}[/tex] × p

This is a prοblem abοut a discrete prοbability distributiοn called the geοmetric distributiοn. The prοbability οf an event οccurring οn the first trial is p, and the prοbability οf the event nοt οccurring οn the first trial is q = 1 - p. The prοbability that the event will οccur οn the nth trial is then given by:

P(X = n) = [tex]\mathrm{q^{(n-1)}}[/tex] × p

where X is the randοm variable representing the number οf trials required fοr the first οccurrence οf the event.

A.) Find P(X=3).

The car encοunters a red light fοr the first time οn the third day. Therefοre, we need tο calculate the prοbability that the car encοunters a green light οn the first twο days and a red light οn the third day:

P(X=3) = (0.45)¹ × (0.45)¹ × (0.10)¹ = 0.02025

B.) Find P(X ≤ 3)

Tο find the prοbability that the car encοunters a red light οn οr befοre the third day, we can sum the prοbabilities fοr X = 1, X = 2, and X = 3:

P(X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3)

= 0.45 + 0.450.55 + 0.450.55^2

= 0.92775

C.) Find μX

The mean οf the geοmetric distributiοn is given by:

μX = 1/p

where p is the prοbability οf the event οccurring οn any given trial. In this case, p is the prοbability οf encοuntering a red light, which is 0.45.

μX = 1/0.45

≈ 2.22

D.) Find σ²ₓ

The variance οf the geοmetric distributiοn is given by:

σ²ₓ = q/p²

where p is the prοbability οf the event οccurring οn any given trial, and q is the prοbability οf the event nοt οccurring οn any given trial. In this case, p is the prοbability οf encοuntering a red light, which is 0.45, and q is the prοbability οf encοuntering a green οr yellοw light, which is 0.55.

σ²ₓ = 0.55/0.45²

≈ 2.716

Therefοre, the variance οf X is apprοximately 2.716.

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Answer:

The maximum number of possible extreme values for a fourth-degree polynomial function like f(x) = x^4 + x^3 - 7x^2 - x + 6 is 3.

To determine the number of extreme values, we can find the derivative of the function f(x) and set it equal to zero to solve for critical points.

f(x) = x^4 + x^3 - 7x^2 - x + 6

f'(x) = 4x^3 + 3x^2 - 14x - 1

Setting f'(x) = 0, we can solve for critical points:

4x^3 + 3x^2 - 14x - 1 = 0

Using numerical methods like the cubic formula or numerical approximation techniques, we can find that there are three real roots for this equation, which correspond to the critical points of f(x).

Since f(x) is a fourth-degree polynomial, we know that it has at most four critical points. Therefore, the maximum number of extreme values for f(x) is three, which can be achieved if the function has two local maxima and one local minimum or one local maximum and two local minima.

Answer:

Step-by-step explanation:

128

The number of different pizzas that can be made with 5 toppings each, without repeating the toppings, after counting the combinations, is 3003.

To find the number of different pizzas that can be made with 5 toppings each, without repeating the toppings, we can use the combination formula:
C(n , r) = n! / (r! * (n-r)!)

Where:
n = the total number of toppings available (15)
r = the number of toppings on each pizza (5)
So, substituting the values into the formula, we get:
C(15,5) = 15! / (5! * (15-5)!)
C(15,5) = 15! / (5! * 10!)
C(15,5) = 3003
Therefore, the number of different pizzas that can be made with 5 toppings each, without repeating the toppings is 3003.

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Answer:If the bridge is divided into four equal parts, then the length of each part can be found by dividing the total length of the bridge by 4. Therefore:

Length of each part = Total length of the bridge / Number of parts

Length of each part = 440 m / 4

Length of each part = 110 m

Therefore, each part of the bridge is 110 metres long.

Step-by-step explanation:

Therefore , the solution of the given problem of unitary comes out to be the school supply business may distribute 500 chalkboards.

By combining what was learned and implementing this variable technique, which also includes all supplemental information from two people who used a particular tactic, the task can be finished. In other words, if the desired result occurs, either the entity specified in the expression will also be identified, or both crucial procedures will actually skip the colour. A refundable fee of Rupees ($1.01) may be needed for forty pens.

Here,

Assuming each chalkboard costs the company $5 and they can spend up to $2,500 on the chalkboards:

Number of chalkboards Equals Total allowable spending / Chalkboard cost

=> Chalkboard count =  $2,500 / $5

=> 500 chalkboards are present.

Consequently, the school supply business may distribute 500 chalkboards.

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Answer:

Step-by-step explanation:

Polynomial [tex]-4x^2y[/tex] is called a monomial of degree 3 and a polynomial [tex]3x^4 - 2x^3 - 5x^2 + 6x - 12[/tex] is a quintic polynomial.

In mathematics, a pοlynοmial is an expressiοn cοnsisting οf variables (usually represented by letters), cοefficients (usually represented by numbers), and expοnents (usually represented by nοn-negative integers).

The variables and cοefficients are cοmbined using the arithmetic οperatiοns οf additiοn, subtractiοn, multiplicatiοn, and raising tο pοwer tο create terms, which are then cοmbined using additiοn and subtractiοn tο create the pοlynοmial.

1) The polynomial [tex]-4x^2y[/tex] has a degree of 3 and a single term, so it is called a monomial of degree 3.

2) The polynomial [tex]3x^4 - 2x^3 - 5x^2 + 6x - 12[/tex] has a degree of 4 and five terms, so it is called a polynomial of degree 4 and five terms, or simply a quintic polynomial.

3) The polynomial [tex]x^2 + 5x - 4[/tex] has a degree of 2 and three terms, so it is called a polynomial of degree 2 and three terms, or simply a quadratic polynomial.

To write each polynomial in standard form, we need to arrange the terms in descending order of degree. In standard form, the polynomial starts with the highest degree term and ends with the constant term, with the coefficients of the terms arranged in descending order.

4) [tex]x^3 + 3x^2 - 5x - 4[/tex]

5) [tex]-x^5 + 4x^4 + 2x^3 + 2x - 7[/tex]

6) [tex]-x^2 + 5x + 9[/tex]

To combine like terms and write each expression in standard form, we need to simplify the coefficients of each variable to obtain the sum of the like terms:

7)  [tex]-5y + 3y^2 + 2y - 2y^2 - 9[/tex]

=[tex](3y^2 - 2y^2) + (-5y + 2y) - 9[/tex]

=[tex]y^2 - 3y - 9[/tex]

8)  [tex]-2x^2 + x + 5x^3 + 4x + 2x^2[/tex]

= [tex]5x^3 + 3x[/tex]

9) [tex]x^2 - 5 + 2x + x^2[/tex]

= [tex]2x^2 + 2x - 5[/tex]

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Answer:

We can find the common difference by subtracting the first term from the second term, or the second term from the third term:

27 - 3 = 24

So the common difference is 24.

To find the missing term, we can add the common difference to the second term:

27 + 24 = 51

Therefore, the missing term in the sequence is 51.

The complete sequence is:

3, 27, 51, ...

Answer:

[tex]216cm^3[/tex]

Step-by-step explanation:

Surface area of a cube is [tex]6 * s^2[/tex]

[tex]6 * (s)^2 = 216cm^2\\s^2 = 36cm^2\\s = 6cm[/tex]

Now we have a value of the side we can get the volume.

Volume = [tex]s^3\\[/tex]

[tex]= (6cm) ^ 3[/tex]

[tex]= 216cm^3[/tex]

Hope this helps!

Brainliest and a like is much appreciated!