Write a polynomial of least degree with real coefficients and with the root -20-7i. Write your answer using the variable x and in standard form with a leading coefficient of 1.

Answer:

Step-by-step explanation:

Farmer Ed has 1,000 meters of​ fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does not fence the side along the​ river, what is the largest area that can be​ enclosed?

Answer:

The number of hamburgers is 136.

Step-by-step explanation:

What we know:

We got cheeseburgers and hamburgers

The total: 408

The number of cheese burgers was 2 times the number of hamburgers

This is pretty much all the info we need to know.

Lets define our variables:

x = hamburgers

y = cheeseburgers

Now, we need our two equations to make a system.

[tex]x+y=408[/tex]

[tex]y=2x[/tex]

To find hamburgers, x, we plug the value of y into our first equation.

[tex]x+2x=408[/tex]

Add like terms

[tex]3x=408[/tex]

Divide by 3

[tex]x=136[/tex]

The number of hamburgers is 136.

Just to check our work, lets find the cheeseburgers.

[tex]y=2(136)[/tex]

[tex]y=272[/tex]

[tex]272+136=408[/tex]

Answer:

f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x.

Step-by-step explanation:

If a polynomial has a complex root, then its conjugate is also a root. Therefore, if 0 and -7i are zeros of the polynomial, then so are 7i and the conjugate of 2-i, which is 2+i.

The factors of the polynomial are then:

(x-0)(x-7i)(x+7i)(x-(2-i))(x-(2+i))

Simplifying the factors, we get:

x(x^2+49)(x^2-4x+5)

Expanding the last factor, we get:

x^5 - 4x^4 + 5x^3 + 49x^3 - 196x^2 + 245x

Finally, we have the 5th degree polynomial:

f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x

Therefore, the polynomial with zeros 0, -7i, and 2-i and 5th degree with real coefficients is f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x.

The solution is, f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x, is  a 5th degree polynomial with real coefficients that has zeros 0, -7i, and 2-i.

Polynomials are sums of k-xⁿ terms, where k can be any number and n can be any positive integer.

here, we have,

If a polynomial has a complex root, then its conjugate is also a root. Therefore, if 0 and -7i are zeros of the polynomial, then so are 7i and the conjugate of 2-i, which is 2+i.

The factors of the polynomial are then:

(x-0)(x-7i)(x+7i)(x-(2-i))(x-(2+i))

Simplifying the factors, we get:

x(x^2+49)(x^2-4x+5)

Expanding the last factor, we get:

x^5 - 4x^4 + 5x^3 + 49x^3 - 196x^2 + 245x

Finally, we have the 5th degree polynomial:

f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x

Therefore, the polynomial with zeros 0, -7i, and 2-i and 5th degree with real coefficients is f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x.

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Answer:

The best option that describes a sample that enables you to draw accurate inferences from the sample is option C: "The sample is random and representative, but not biased."

A random sample is one in which every member of the population has an equal chance of being selected. This helps to ensure that the sample is not influenced by any outside factors that could skew the results.

A representative sample is one that accurately reflects the characteristics of the population it is drawn from. This ensures that the results obtained from the sample can be generalized to the larger population.

However, if the sample is biased, it may not accurately represent the population. This could lead to inaccurate inferences and conclusions.

Therefore, a random and representative sample that is not biased is the best option for drawing accurate inferences from the sample.

Step-by-step explanation:

Here the option is (D): Sum of Cubes.

The Given expression can be written as:

a³ + b³ = (a + b)(a² - ab + b²)

we can write 8x³ as (2x)³ and 243 as 3³.

we have,

8x³ + 243 = (2x)³ + 3³

we can rewrite the above equation

(2x + 3)(4x² - 6x + 9)

Therefore, the identity that can be used to rewrite the given expression is an option (D): Sum of Cubes

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There are 20 different election outcomes are possible.

An outcome is a possible result of a random experiment or event. It is a specific occurrence that may or may not happen, but that could happen under certain conditions. Outcomes are often associated with probabilities, which are measures of the likelihood that a particular outcome will occur.

This is a problem of counting the number of ways to choose 3 people out of 6 for 3 distinct positions. We can solve this using the combination formula:

ⁿCr = C(n, r) = n! / (r! * (n-r)!)

C(6, 3) = 6! / (3! * (6-3)!) = 20

Therefore, there are 20 different election outcomes possible for the president, secretary, and treasurer positions among a group of 6 people.

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Penny plans to distribute a questionnaire at the entrance to the coffee shopping center in order to do market research because she would love to build a coffee shop there which are explained.

Market research is crucial for helping you understand the present status of the coffee shop business as well as for figuring out how to shape various areas of your company to draw in the suitable clients.

Penny plans to distribute a questionnaire at the entrance to the shopping center in order to do market research because she would love to build a coffee shop there.

Penny may utilize a questionnaire to collect the data she needs, which would include the following:

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The linear inequality in open-sentence form are

An open sentence is a mathematical statement that contains at least one blank or unknown and that becomes true or false when the blank is filled or a quantity is substituted for the unknown.

Now, since we have the expressions,

First, we note that there are 3 open sentences and only two of them are inequalities.

The 3 open sentences are

The two inequalities are

So, the linear inequality in open-sentence form are

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Answer:

x = log(3) (8)

or log 8 base 3

Step-by-step explanation:

some number^x = another number

some number = base (the lower case number)

another number = the big number next to it.

put log infront in these situations.

Answer:71

Step-by-step explanation:

-3-(-10)-(-64)=-3+10+64=71

Thus, the length of the missing side for the given Right triangles using  the Trigonometric Ratios is found as :x = 18.47 units.

The leg, denoted as, that is next to the 30 degree angle θ is represented by Side BC. To get the length of side BD, the hypotenuse of triangle BCD, we use the cosine ratio.

cos θ = Adjacent leg / Hypotenuse = BC/BD

cos 30° = 8√2 / BD

Now, we know.

cos 30° = √3/2

√3/2 =  8√2 / BD

BD = 8√2 / √3/2

BD = 16√2 / √3

On rationalising;

BD = 16√2*√3 / √3*√3

BD = 16√6 / 3

Sin A = Opposite leg /Hypotenuse = BD / x

Sin 45° = (16√6 / 3) / x =  16√6 / 3x

We know that:

Sin 45° = 1/√2

(1/√2) = 16√6 / 3x

(1/√2) * x = 16√6 /3

Or,

√2/2 * x = (16√6 /3) * (2/√2)

x = 32√3 / 3

x = 18.47

Thus, the length of the missing side for the given Right triangles using  the Trigonometric Ratios  is found as :x = 18.47 units.

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The correct question is-

Find the missing side length. Give the answer as radicals in simplest form.

Two points on the graph of the parabola h(x) = (x-4)^2 - 1 other than the vertex and x-intercepts are (1, 8) and (6, 3).

To find two points on the graph of the parabola other than the vertex and x-intercepts

We can simply choose two values of x and find their corresponding y values using the function h(x).

Let's choose x = 1:

h(1) = (1-4)^2 - 1 = 9 - 1 = 8

So one point on the graph is (1, 8).

Now let's choose x = 6:

h(6) = (6-4)^2 - 1 = 4 - 1 = 3

So another point on the graph is (6, 3).

Hence, the two points are (1, 8) and (6, 3)

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The dataset is evenly distributed, so the appropriate measure of variability is the range

The dataset is evenly has no outliers, so the appropriate measure of center is the mean

Measure of variability

When the data set is evenly distributed, a measure of variability appropriate for stem and leaf plot would be the range

The range is simply the difference between the largest and smallest value in the data set.

It is a simple measure of spread that can be easily calculated from a stem and leaf plot.

Measure of center

When there are no outliers in a dataset, the appropriate measure of center to use in a stem and leaf plot is the mean

The mean, also known as the average, is calculated by summing all the values in the dataset and then dividing by the total number of values.

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Answer:

Step-by-step explanation:

SEE THE GRAPH

The domain of the function f(x)=7x+5/3x-1 is all real numbers except x = 1/3. This means that the function will produce a real number for any value of x except for x = 1/3.

Function in mathematics is a relation which assigns each element of a set (called the domain) to exactly one element of another set (called the codomain). It can also be thought of as a type of mapping from one set to another. Functions are used to model real-world problems and to simplify the expression of a certain problem. They are widely used in algebra, trigonometry and other areas of mathematics.

The domain of a function is the set of all possible inputs for the function. In other to determine the domain of a function, we must first identify the independent variable (x) and then determine which values of x can be used in the function.

In the given function, f(x)=7x+5/3x-1, the independent variable is x. To find the domain of this function, we must identify the values of x that will produce results that are real numbers. To do this, we must set the denominator (3x-1) equal to zero and solve for x.

We can solve this equation by factoring and setting each factor equal to zero:

3x - 1 = 0

3x = 1

x = 1/3

Therefore, the domain of the function f(x)=7x+5/3x-1 is all real numbers except x = 1/3. This means that the function will produce a real number for any value of x except for x = 1/3.

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For the graph of the rational function the as x → −∞​, f(x) → −∞​.

A rational function is the ratio of two polynomial functions with a non-zero denominator polynomial. R(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions, is the standard representation of this relationship. We studied the rational number idea in previous grades. When the denominator is not equal to zero, it is the quotient or ratio of two integers. Rational, then, is a name that comes from the term ratio.

From the given graph we observe that as the value of x tends to - ∞, the value of f(x), that is, the value of y constantly goes upwards and leftwards.

Thus,

As x → −∞​, f(x) → −∞​.

Hence, for the graph of the rational function the as x → −∞​, f(x) → −∞​.

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Answer:

Translation of three units to the right (on the x-axis) and one unit down (on the y-axis).

If you want, this can be represented in vector form (3 1).

Step-by-step explanation:

This was done by translating the shape three units to the right and one unit down

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The equation of the line passing through the points (0, 7) and (2, -5) is y = -6x + 7

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

The standard form of a linear equation is:

y = mx + b

where m is the slope(rate of change) and b is the y intercept

As shown in the graph, the line passes through the points (0, 7) and (2, -5). Hence:

[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1} (x-x_1)\\\\substituting:\\\\y-7=\frac{-5-7}{2-0} (x-0)\\\\y=-6x+7[/tex]

The equation of the line is y = -6x + 7

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The value of the 5 measurements according to the given function is 66.

When a group of numbers, known as addends or summands, are added together in mathematics, the outcome is their sum or total. Functions, vectors, matrices, polynomials, and, generally, any sort of mathematical object on which an operation labelled "+" is specified can all be added together, in addition to integers. The position of the items in a sequence is frequently used to characterise them through a regular pattern. Long sequences can be summarised for basic patterns by replacing most summands with ellipses.

The function is given as:

[tex]\sum_{i = 1}^5 (x_i^2)\\\\\sum_{i = 1}^5 = x_1 + x_2 + x_3 +x_4 + x_5\\\\\sum_{i = 1}^5 (x_i^2) = 12 + 14 + 16 + 7 + 17 = 66[/tex]

Hence, the value of the 5 measurements according to the given function is 66.

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The firm's profit in this case is = $360.

What is the total cost?

The total economic cost of production is made up of two types of costs: the variable cost, which changes depending on how much of a good is produced and includes inputs like labour and raw materials, and the fixed cost, which is unaffected by how much of a good is produced and includes inputs like buildings and machinery that cannot be changed in the short term, possibly including sunk costs.

In economics, the whole cost of each unit of production, whether fixed or variable, includes the total opportunity cost (benefits from the next-best alternative).

Example:

Q        Fixed       Variable    Total    Marginal    Aver.     Aver.     Aver.

       Costs        Costs         Cost    Cost           FC         VC         TC      

0          100             0              100         -               -             -             -

1           100           50               150       150           100         50         150

2          100           70               170         20            50         35          85

3          100           90               190        20           33.33      30        63.33

4          100          140               240       50            25          35          60

5          100         200               300       60            20          40          60

6          100         360              460      160           16.67       60        76.67

The firm's profit in this case is = $360.

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Answer: -5/6

Step-by-step explanation:

Hopefully this helps, my friend, best of luck to you :)

Answer:

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The values of h1 and h2 for which v3 is a linear combination of v1 and v2 is given as follows:

Vector v3 being a linear combination of v1 and v2 means that the following system of equations is used to relate the three vectors:

xv1 + yv2 = v3.

From each line of each vector, the system of equations relating these amounts is given as follows:

From the second equation, considering h1 = 1, the value of y is given as follows:

1 = -y

y = -1.

Hence, from the third equation, the value of h2 is obtained as follows:

-1 + 3(-1) = h2

-1 - 3 = h2

h2 = -4.

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If we take the values are h₁ = -1,  h₂ = 2, then the last vector is a linaer combination of the first two.

Here we want to find the values of h₂ and h₃ such that the last vector is a linear combination of the first one, that means that we can write:

a*v₁ + b*v₂ = v₃

For each of the components we can write:

a*1 + b*0 = 1

a*0 + b*-1 = h₁

a*-1 + b*3 = h₂

With the first equation we conclude that a = 1, then:

1*0 + b*-1 = h₁

1*-1 + b*3 = h₂

If we also take b = 1 we will get:

0 - 1= h₂

-1 + 3 = h₃

Then the values are h₁= -1,  h₂ = 2.

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The numeric value of f(x) = 3x² - 7x - 4 at x = 0 is given as follows:

f(0) = -4.

To obtain the numeric value of a function or of an expression, we substitute each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

The function for this problem is defined as follows:

f(x) = 3x² - 7x - 4

We want to find the numeric value of the function at x = 0, hence both instances of x in the definition of the function are replaced by zero and then we add the terms, as follows:

f(0) = 3(0)² - 7(0) - 4

f(0) = -4.

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Therefore , the solution of the given problem of expressions comes out to be b + 8 when 8a-2b+8 and -2a+6b are added together.

There is a need for calculations like joining, random variable division, variable multiplication, and removal. If they were united, the following would happen: A mathematical equation, some data, and a formula. A statement of truth is composed of numerical data, formulae, components, and mathematical operations like additions, subtractions, errors, and segmentations. One can assess and analyse words and sentences.

Here,

The formula is as follows:

=> 8a - 2b + 8 - 2a + 6b - (5b - 6a)

By combining similar words to make the expression shorter, we get:

=> (0a - b + 8 = -b + 8)

=>  (8a - 2a - 6a) + (-2b + 6b - 5b)

Therefore, the answer is b + 8 when 8a-2b+8 and -2a+6b are added together.

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b) The mean and standard deviatiοn οf the data is 8.022 and 0.335 respectively.

In statistics, mean is measure οf central tendency that represents average value οf set οf numbers. It is calculated by adding up all  values in  set and dividing  sum by the tοtal number οf values in  set.

b) Tο determine the mean and standard deviatiοn οf the data, we can input the given values intο a calculatοr οr a spreadsheet prοgram and use the apprοpriate fοrmulas. Here are the calculatiοns:

Mean = (7.96 + 8.54 + 7.92 + 8.40 + 8.02 + 7.91 + 8.12 + 7.85 + 8.27 + 7.67) / 10

= 8.022

Standard deviatiοn = sqrt([(7.96 - 8.022)² + (8.54 - 8.022)² + ... + (7.67 - 8.022)²] / 9)

= 0.335 (rοunded tο 3 decimal places)

Therefοre, the mean petrοl price is 8.022 and the standard deviatiοn is 0.335.

c) The histοgram appears tο be slightly skewed tο the right, as the tail οf the histοgram extends further tο the right than tο the left.

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Answer:

The transformation that converts the graph of f(x)=2|x|+4 into the graph of g(x)=|x|+2 is a Vertical Shrink.

To see this, note that the 2 in f(x) causes a vertical stretch of the absolute value graph. Removing the 2 in g(x) results in a Vertical Shrink of the absolute value graph. The +4 and +2 only cause a vertical shift of the graph.

Answer: Vertical Shrink

Step-by-step explanation:

The transformation that converts the graph of f(x)=2|x|+4 into the graph of g(x)=|x|+2 is a vertical shrink.

To see why, let's examine each function separately:

Therefore, the transformation from f(x) to g(x) involves removing the vertical stretch by dividing the absolute value term by 2. This results in a smaller "V" shape for the graph of g(x), or a vertical shrink.

The answer is:

[tex]$\boxed{\text{Vertical Shrink}}$[/tex]

The revised questionnaire that is more suitable that includes reasons for taking a GAP year is given below:

The new questionnaire includes four questions aimed at gaining insight into the participant's motivations, goals, and timeframe for taking a gap year:

Have you considered taking a gap year before starting your next academic or professional endeavor?

What are your top 5 reasons for considering a gap year? (Please list in order of importance)

What are your goals for your gap year? (e.g., travel, volunteer work, internships, personal development, etc.)

How long do you plan to take for your gap year?

Reasons for taking a gap year may vary from person to person, but here are five common motivations for doing so:

To gain life experience and personal growth

To travel and explore new cultures

To gain work experience or internships in a desired field

To take a break from academic or professional responsibilities

To pursue passions or hobbies that may not be feasible during a regular academic or work schedule.

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Revise the suggested questionnaire so that it is more suitable. Include 5 reasons for taking a GAP YEAR. Submit a copy of a new revised questionnaire

Have you considered taking a gap year before starting your next academic or professional endeavor?

What are your reasons for considering a gap year?

What are your goals for your gap year?

How long do you plan to take for your gap year?

Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer