can you help me with this task please? thank you!!

The polynomial function of the least degree with integral coefficients that has the zeros [tex]1 + 3i[/tex], [tex]-2i[/tex] is f(x) = x⁴ - 2x³ + 14x² - 8x + 40.

From the question, we are to write a polynomial function of the least degree with integral coefficients that have the given zeros.

The given zeros are [tex]1+3i[/tex],[tex]-2i[/tex].

If [tex]1 + 3i[/tex] and [tex]-2i[/tex]are zeros of a polynomial function with integral coefficients, then their conjugates [tex]1 - 3i[/tex] and [tex]2i[/tex] are also zeros of the function.

To find the polynomial function, we can use the fact that if r is a zero of a polynomial function, then (x - r) is a factor of the function. Thus, we can start by writing out the factors corresponding to each of the zeros:

[tex](x - (1 + 3i))(x - (1 - 3i))(x - (-2i))(x - 2i)[/tex]

Next, we can simplify these factors by multiplying them out:

[tex][(x - 1) - 3i][(x - 1) + 3i](x + 2i)(x - 2i)\\= [(x - 1)^2 - (3i)^2](x^2 - (2i)^2)[/tex]

= [(x - 1)² + 9](x² + 4)

Expanding the terms, we get:

(x² - 2x + 10)(x² + 4)

Multiplying out the factors, we obtain:

x⁴ - 2x³ + 10x² + 4x² - 8x + 40

Simplifying this expression, we get:

x⁴ - 2x³ + 14x² - 8x + 40

Hence, the polynomial function is f(x) = x⁴ - 2x³ + 14x² - 8x + 40

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Jane sold 27 boxes of cookies last year if Jane sold 50 boxes of cookies this year. That is 4 less than 2 times the number of boxes she sold last year.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

Given ,

Jane sold 50 boxes of cookies this year. That is 4 less than 2 times the number of boxes she sold last year.

Let's assume the number of boxes Jane sold last year as x.

As per the given condition, 50 = 2x - 4

Adding 4 on both sides, we get:

50 + 4 = 2x - 4 + 4

54 = 2x

Dividing by 2 on both sides, we get:

x = 27

Therefore, Jane sold 27 boxes of cookies last year.

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For the sample size of 20, the 95% confidence interval is (-4.87, 27.16), which is narrower than the previous interval due to the larger sample size.

In statistics, mean is a measure of central tendency that represents the average value of a set of numbers. It is calculated by summing up all the numbers in the set and dividing the sum by the total number of values in the set. Mean is also commonly referred to as the arithmetic mean. It is a commonly used statistical measure in many fields including finance, economics, social sciences, and more.

Here,

To calculate the z-scores associated with each student, we first need to calculate the sample mean and standard deviation for the Distance to Work variable:

Sample mean: (0+0+5+10+15+30+32)/7 = 10.71

Sample standard deviation: √(((0-10.71)² + (0-10.71)² + (5-10.71)² + (10-10.71)² + (15-10.71)² + (30-10.71)² + (32-10.71)²)/6) = 10.72

Now we can calculate the z-scores for each student:

Student 1: (0 - 10.71) / 10.72 = -0.94

Student 2: (0 - 10.71) / 10.72 = -0.94

Student 3: (5 - 10.71) / 10.72 = -0.53

Student 4: (10 - 10.71) / 10.72 = -0.07

Student 5: (15 - 10.71) / 10.72 = 0.40

Student 6: (30 - 10.71) / 10.72 = 1.80

Student 7: (32 - 10.71) / 10.72 = 1.98

Student 8: (8 - 10.71) / 10.72 = -0.25

Student 9: (36 - 10.71) / 10.72 = 2.37

Student 10: (9 - 10.71) / 10.72 = -0.16

Student 11: (22 - 10.71) / 10.72 = 1.05

Student 12: (10 - 10.71) / 10.72 = -0.07

Student 13: (15 - 10.71) / 10.72 = 0.40

Student 14: (28 - 10.71) / 10.72 = 1.54

Student 15: (12 - 10.71) / 10.72 = 0.12

Student 16: (6 - 10.71) / 10.72 = -0.44

Student 17: (0 - 10.71) / 10.72 = -0.94

Student 18: (10 - 10.71) / 10.72 = -0.07

Student 19: (30 - 10.71) / 10.72 = 1.80

Student 20: (0 - 10.71) / 10.72 = -0.94

To identify potential outliers, we can look for z-scores that are more than 2 standard deviations away from the mean (i.e., greater than 2 or less than -2). From the list above, we can see that Students 6, 7, 9, and 19 have z-scores greater than 2, indicating that they may be potential outliers.

For the second sample of seven data points (0, 0, 5, 10, 15, 30, 32), the mean is 11.14 and the standard deviation is 12.05. Using a t-distribution with 6 degrees of freedom (n-1), the 95% confidence interval is (-7.54, 29.82).

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The first step in developing a strategy for investing in two different stocks is to perform research on the stocks and their respective industries.

Stock is a type of security that signifies ownership in a company and represents a claim on part of the company's assets and earnings. It is a share of ownership in a company. It is bought and sold on a stock exchange, and the price of the stock is determined by the supply and demand of the stock.

This can be done by looking at the company’s financials, track record, industry trends, and more. This research can help you to determine which stock is more likely to perform better in any given economic condition. By doing this, you can maximize your return while also minimizing your risk. This can be done by using a risk-adjusted return metric such as the Sharpe Ratio.

Finally, you must decide how to allocate funds across the two stocks. This will depend on your risk tolerance and investment goals. For example, if you are a conservative investor, you may want to allocate more funds to the stock with the highest expected return under all economic conditions. On the other hand, if you are a more aggressive investor, you may want to allocate more funds to the stock with the highest expected return under one or two of the economic conditions.

Overall, developing a strategy for investing in two different stocks takes into account several factors, including research, expected returns, and risk tolerance. By doing this, you can maximize your return while also minimizing your risk.

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In the given demand equation, the value of x and corresponding price p that maximize revenue are x = 31.8 and p = $514.60.

To find the value of x and corresponding price p that maximize revenue, we first need to determine the revenue function. Revenue (R) is given by the product of price (p) and quantity demanded (x), so:

R(x) = xp

Substituting the given demand equation for p, we get:

R(x) = x(1/2 + ² - 20x + 1200)

Simplifying and rearranging:

R(x) = 1/2x + ²x - 20x² + 1200x

R(x) = -20x² + (1200 + ²)x + 1/2x

To find the value of x that maximizes revenue, we take the derivative of R(x) with respect to x, set it equal to zero, and solve for x:

dR/dx = -40x + (1200 + ²) + 1/2 = 0

Solving for x:

x = (1200 + ² + 1/2) / 40

x = 31.8

To find the corresponding price, we substitute this value of x back into the demand equation for p:

p = 1/2 + ² - 20(31.8) + 1200

p = $514.60

Therefore, the value of x and corresponding price p that maximize revenue are x = 31.8 and p = $514.60.

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Answer:

0.19^9

0.19 to the 9th power

Step-by-step explanation:

Use Exponent Rules:

If you are multiplying terms with the same base, then ADD the exponents.

see image

Answer:

5 < x < 21

Step-by-step explanation:

given 2 sides of a triangle then the third side x is in the range

difference of 2 sides < x < sum of 2 sides , that is

13 - 8 < x < 13 + 8 , then

5 < x < 21

The conclusion is that if A is true, B is true, then C must be true as well.

What is statement ?

In logic and mathematics, a statement is a declarative sentence that is either true or false. It is an assertion about a fact or a relationship between facts. A statement can be a simple sentence, such as "The sun is shining," or a complex one, such as "If it rains tomorrow, I will stay indoors."

If the statement "If A AND B, then C" is true, and if its reverse statement is also true, then the converse of the statement is "If C, then A AND B."

If A is true and B is true, then A AND B is also true. Therefore, by the original statement, C must also be true.

Therefore, the conclusion is that if A is true, B is true, then C must be true as well.

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Answer:

[tex]\mathrm{y\:=\:\frac{3}{2}x\:+\:7}[/tex]

Step-by-step explanation:

To find the equation of a line parallel to -4y = 32 - 6x, we first need to rewrite this equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

-4y = 32 - 6x

Divide both sides by -4:

y = -8 + (3/2)x

So the slope of this line is 3/2. Since we want a line parallel to this one, it will have the same slope of 3/2.

Now we can use the point-slope form of a linear equation to find the equation of the line that passes through (0,7) with a slope of 3/2:

y - y1 = m(x - x1)

where m = 3/2 and (x1,y1) = (0,7)

y - 7 = (3/2)(x - 0)

Simplifying:

y - 7 = (3/2)x

y = (3/2)x + 7

So the equation of the line parallel to -4y = 32 - 6x that passes through the point (0,7) is y = (3/2)x + 7.

Hello and Regards zoecenters91522.

Therefore, the distance between the points (15,-17) and (-10,-17) is 25 units.

The distance between two points is the measure of the length of the straight line that joins them. On a Cartesian plane, the distance between two points is calculated using the Euclidean distance formula, which states that the distance between two points (x₁, y₁) and (x₂, y₂) is equal to the square root of the sum of the squares of the differences of their coordinates:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

This formula can be used to calculate the distance between two points on any plane, such as a three-dimensional plane or Euclidean space of any dimension.

Now we locate the points of x and y.

We substitute the points in the formula and solve:

[tex]\huge \boxed{d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }[/tex]

[tex]\huge \boxed{d=\sqrt{(-10-15)^2+(-17-(-17))^2 } }[/tex]

[tex]\huge \boxed{d=\sqrt{(-25)^2+0^2}=\sqrt{625+0} }[/tex]

[tex]\huge \boxed{d=\sqrt{625}=25 \ units }[/tex]

Therefore, the distance between the points (15,-17) and (-10,-17) is 25 units.

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Answer:

7/8 - 5/7 =

49-40/56 =

9/56

Answer:

answer is B

Step-by-step explanation:

-4x-7x=-11x

+3-8=-5

=-11x-5

answer: B -11x-5

[tex] \sf \implies \: - 4x +3 - 7x - 8[/tex]

[tex] \sf \implies \: - 11x +3 - 8[/tex]

[tex] \sf \implies \: - 11x - 5[/tex]

Hence, Option B is correct!!

The solutions to the indicated quantities are x = 25, MIK = 25 and KIE = 50 degrees

The value of x

From the question, we have the following angle that can be used in our computation:

Angle MIE = 75

Using the above as a guide, we have the following:

x + 2x = 75

Evaluate

3x = 75

So, we have

x = 25

The angle MIK

From the figure, we have

MIK = x

So, we have

MIK = 1 * 25

MIK = 25

The angle KIE

From the figure, we have

KIE = 2x

So, we have

KIE = 2 * 25

KIE = 50

Hence, the measure of angle KIE is 50 degrees

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The difference between the interest paid with the minimum monthly payment and the interest paid with the $527.16 monthly payment is $83,421.

What is the interest?

Interest is the price you pay to borrow money or the cost you charge to lend money. Interest is most often reflected as an annual percentage of the amount of a loan. This percentage is known as the interest rate on the loan.

First, we need to calculate the monthly payment at a 10% interest rate for 25 years:

PV = 90,000

i = 0.10/12

n = 25*12 = 300

Using the formula for the monthly payment of a mortgage loan, we get:

[tex]P = i PV / (1 - (1 + i)^{(-n)})[/tex]

= 805.23

So, if Jackie paid the minimum monthly payment, she would pay $805.23 per month. The total amount paid over the lifetime of the loan would be:

Total amount paid = P * n = 805.23 * 300 = $241,569

The amount of interest paid would be:

Interest paid = Total amount paid - PV = 241,569 - 90,000 = $151,569

If Jackie had made payments of $527.16 per month, the total amount paid over the lifetime of the loan would be:

Total amount paid = 527.16 * 300 = $158,148

The amount of interest paid would be:

Interest paid = Total amount paid - PV = 158,148 - 90,000 = $68,148

Therefore, the difference between the interest paid with the minimum monthly payment and the interest paid with the $527.16 monthly payment is: $151,569 - $68,148 = $83,421

hence, the answer is not one of the choices provided.

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Answer:

radius r = 22.1 m

height h = 5.91 m

slant height s = 22.8765841 m

volume V = 3022.73898 m^3

lateral surface area L = 1588.30288 m^2

base surface area B = 1534.38527 m^2

total surface area A = 3122.68815 m^2

Step-by-step explanation:

Answer:

The total number of batteries that Emily buys can be expressed as:

8p + 30b

Here, 8p represents the total number of batteries in the packs that Emily buys, and 30b represents the total number of batteries in the boxes that Emily buys. By adding these two terms together, we can find the total number of batteries that Emily buys.

Answer:

Opposite (or a any synonyms)
Addition (Or sum)
division
inverse

Step-by-step explanation:

The first one is the most complicated, theres so many words can fit. Probably try with an synonyms of reverse like "Opposite"  
For the 2nd,3rd and 4th options:

Addition (Or sum) since is the reverse of subtraction
division  since is the reverse of multiplication
inverse since those functions (Exponent and radicals) are also inverse of each other

Answer:

To estimate the dealership's cost for the car, we need to first calculate the dealership's selling price. We know that the car has an MSRP of $34,995.99, and we know that dealerships typically have a 20% markup on selling price.

So, to find the dealership's selling price, we can use the formula:

Selling price = MSRP + (Markup percentage * MSRP)

Substituting the values we have:

Selling price = $34,995.99 + (20% * $34,995.99)

Selling price = $34,995.99 + $6,999.20

Selling price = $41,995.19

Therefore, we estimate that the dealership paid $41,995.19 - the 20% markup - for the car.

To check this estimate, we can calculate the dealership's cost using the formula:

Cost = Selling price - Markup

Substituting the values we have:

Cost = $41,995.19 - (20% * $41,995.19)

Cost = $41,995.19 - $8,399.04

Cost = $33,596.15

This is very close to the MSRP of $34,995.99, which suggests that our estimate is reasonable.

The height of the tree is equal to 210 ft to the nearest foot using the sine rule.

The sine rule is a relationship between the size of an angle in a triangle and the opposing side.

angle opposite of the tree = 56 - 26 = 30

angle opposite of the shadow = 180 - 90 - 56= 34

by application of sine rule:

sin 30/h = sin 34/235

h = (235 × sin 30)/sin 34 {cross multiplication}

h = 117.5/0.5592

h = 210.1216 ft.

Therefore, the height of the tree is equal to 210 ft to the nearest foot using the sine rule.

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The general sοlutiοn tο the differential equatiοn is [tex]\mathrm{y = Ce^{(sin(x))}}[/tex].

The derivative οf a functiοn represents the instantaneοus rate οf change οf the functiοn at a specific pοint. It is calculated by finding the limit οf the difference quοtient as the interval between twο pοints οn the functiοn apprοaches zerο. The derivative can be expressed as a functiοn οf the independent variable, and it prοvides valuable infοrmatiοn abοut the behaviοr οf the οriginal functiοn.

The prοcess οf differentiatiοn invοlves applying a set οf rules tο functiοns tο οbtain their derivatives. These rules include the pοwer rule, prοduct rule, quοtient rule, chain rule, and οther mοre advanced rules that are used tο differentiate mοre cοmplex functiοns.

Tο sοlve the given differential equatiοn, we can use the methοd οf integrating factοrs.

First, we can rewrite the equatiοn as:

y'cοsx = ysinx + sin(175x)

Next, we can multiply bοth sides by the integrating factοr, which is [tex]e^{(\int(cos(x) dx))} = e^{(\sin(x) + C)}[/tex], where C is a cοnstant οf integratiοn:

[tex]\mathrm {e^{(sin(x)) }y'cosx = e^{(sin(x))} ysinx + e^{(sin(x))}sin(175x) + Ce^{(sin(x))}}[/tex]

Nοw, we can recοgnize the left-hand side as the derivative οf [tex]e^{(sin(x))}y[/tex]:

[tex](e^{(sin(x))y)}' = e^{(sin(x))} y' + cos(x) e^{(sin(x))}y[/tex]

Substituting this intο the abοve equatiοn, we get:

[tex]\mathrm{(e^{(sin(x))y)}' = e^{(sin(x)) }ysinx + e^{(sin(x))}sin(175x) + Ce^{(sin(x))}}[/tex]

[tex]cos(x) e^{(sin(x))}y = e^{(sin(x))y)}'[/tex]

Separating variables and integrating bοth sides, we get:

[tex]\int e^{sin(x) }dy/y = \int cos(x) dx[/tex]

ln|y| + C = sin(x) + C'

where C' is anοther cοnstant οf integratiοn.

Therefοre, the general sοlutiοn tο the differential equatiοn is:

[tex]\mathrm{|y| = e^{(sin(x)) }e^{(C' - sin(x))}}[/tex]

[tex]\mathrm{y = \± e^{(C' - sin(x) + sin(x))}}[/tex]

[tex]\mathrm{y = Ce^{(sin(x))}}[/tex]

where C is an arbitrary cοnstant.

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The proof that the quadrilateral is a rectangle is shown below

From the question, we have the following parameters that can be used in our computation:

A = (-2, 3)

B = (-4, 1)

C = (2, -1)

D = (0, -3)

From the graph, we have the following lengths

AB = √8

CD = √8

BD = √32

AC = √32

The above shows that opposite sides are equal

From the graph, we have the following slopes

AB = 1

CD = 1

BD = -1

AC = -1

The above shows that opposite sides are have equal slopes and adjacent sides are their slopes to be opposite reciprocals

Hence, the quadrilateral is a rectangle

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Answer:

Step-by-step explanation:

Let’s denote the production levels of coal, electricity and steel as x, y and z respectively. We can set up a system of equations to represent the inter-industry demand for each industry’s output.

For coal: 0.02x + 0.04y + 0.10z = x For electricity: 0.30x + 0.04y = y For steel: 0.30x + 0.02y + 0.04z = z

Solving this system of equations gives us x = (50/3)y and z = (25/2)y.

The new sales demand for coal is $2 billion (double the original), for electricity is $3 billion (triple the original) and for steel is $6 billion (an increase of 50%). Substituting these values into our equations gives us:

(50/3)y = $2 billion y = $3 billion (25/2)y = $6 billion

Solving these equations gives us y = $3 billion, x = $5 billion and z = $18.75 billion.

So to satisfy the new demand, the coal industry should produce at a level of $5 billion, the electric industry should produce at a level of $3 billion and the steel industry should produce at a level of $18.75 billion.

Step-by-step explanation:

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

Given the slope m = -2 and the y-intercept (a, b) = (-5, -4), we can substitute these values into the equation to get:

y = mx + b

y = -2x + (-4)

y = -2x - 4

Therefore, the equation in slope-intercept form is y = -2x - 4.

Answer:

The correct answer is: slope= -1;  y-intercept (0,2)

Step-by-step explanation:

The given equation is:   [tex]q=2-r[/tex]

Assuming [tex]q[/tex] and [tex]r[/tex] as [tex]y[/tex] and [tex]x[/tex] respectively, the equation becomes:  [tex]y=2-x[/tex]

Rearranging the terms in right side:  [tex]y=-x+2[/tex]

Comparing this above equation with the slope-intercept form [tex]y=mx+b[/tex], we will get.....

[tex]mx=-x[/tex] ⇒ [tex]m=\frac{-x}{x} =-1[/tex]

[tex]and[/tex]

[tex]b=2[/tex]

So, the slope is -1 and y-intercept is (0, 2)

Answer:

After 15 years, the amount in the account will be $21.611.32, assuming the interest is compounded annually.

Step-by-step explanation:

To calculate the amount after 15 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the amount, P is the principal (initial amount), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).

In this case, P = $12,000, r = 0.04 (4% expressed as a decimal), n = 1 (compounded annually), and t = 15 years.

Plugging in the values, we get:

A = $12,000(1 + 0.04/1)^(1*15)

A = $12,000(1.04)^15

A = $12,000(1.801)

A = $21,611.32

Therefore, after 15 years, the amount in the account will be $21.611.32, assuming the interest is compounded annually.

Answer:

answer is 11.60

Step-by-step explanation:  

formula is C^2 = a^2 + b^2

c = 13

a = 8

b = ?

169 = 64 + x^2

135 = x^2

I hope this helps!

According to this inequality, the number of persons attending a theatrical inequality  play, denoted by p, must be less than or equal to 275 in order for everyone to have a seat.

In mathematics, an inequality is a non-equal connection between two expressions or values. As a result, imbalance leads to inequity. In mathematics, an inequality connects two values that are not equal. Inequality is not the same as equality. When two values are not equal, the not equal symbol is typically used (). Various disparities, no matter how little or huge, are utilised to contrast values. We resuming our current status quo. Yet a variety of things lead to inequality: Negative values on both sides are split or added.

a. The disparity in representing the voting age in the United States is as follows:

a ≥ 18

In order to be eligible to vote, a voter's age, denoted by a, must be more than or equal to 18 years old.

b. The inequality used to depict a theater's seating capacity is:

p ≤ 275

According to this inequality, the number of persons attending a theatrical play, denoted by p, must be less than or equal to 275 in order for everyone to have a seat.

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