What is the solution of
O x≤-3 or 2 Ox<-3 or 2 O-3≤x≤2
or x > 7
O-3 7
x²+x-6
<0?
X-7 50₂

The integral of the function expressed as sum of partial frictions is -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C.

First, factor out 2x from the denominator to obtain:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[(x + 3)/(2x)(x² - 4)] dx

Next, we use partial fractions to express the integrand as a sum of simpler fractions. To do this, we need to factor the denominator of the integrand:

2x(x² - 4) = 2x(x + 2)(x - 2)

Therefore, we can write:

(x + 3)/(2x)(x² - 4) = A/(2x) + B/(x + 2) + C/(x - 2)

Multiplying both sides by the denominator, we get:

x + 3 = A(x + 2)(x - 2) + B(2x)(x - 2) + C(2x)(x + 2)

Now, we need to find the values of A, B, and C. We can do this by equating coefficients of like terms:

x = A(x² - 4) + B(2x² - 4x) + C(2x² + 4x)

x = (A + 2B + 2C)x² + (-4A - 4B + 4C)x - 4A

Equating coefficients of x², x, and the constant term, respectively, we get:

A + 2B + 2C = 0

-4A - 4B + 4C = 1

-4A = 3

Solving for A, B, and C, we find:

A = -3/4

B = 7/16

C = -1/16

Therefore, the partial fraction decomposition is:

(x + 3)/(2x)(x² - 4) = -3/(4(2x)) + 7/(16(x + 2)) - 1/(16(x - 2))

The integral becomes:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

Integrating each term separately gives:

∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

= -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

where;

Therefore, the final answer is:

∫[(x + 3)/(2x³ - 8x)] dx = -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

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Answer:

A large equilateral triangle pyramid stands in front of the city's cultural center. Each side of the base measures 40 feet and the slant height of each lateral side of the pyramid is 50 feet.

A painter can paint 100 square feet of the pyramid in 18 minutes.

How long does it take the painter to paint 75% of the pyramid?

Step-by-step explanation:

The total surface area of the pyramid can be calculated using the formula for the lateral surface area of a pyramid:

Lateral surface area = (1/2) × perimeter of base × slant height

Since the base is an equilateral triangle, the perimeter is 3 times the length of one side:

Perimeter of base = 3 × 40 feet = 120 feet

Lateral surface area = (1/2) × 120 feet × 50 feet = 3000 square feet

To paint 75% of the pyramid, the painter needs to paint:

0.75 × 3000 square feet = 2250 square feet

Since the painter can paint 100 square feet in 18 minutes, the time required to paint 2250 square feet can be calculated as:

2250 square feet ÷ 100 square feet per 18 minutes = 225 ÷ 10 × 18 minutes = 405 minutes

Therefore, the painter would need 405 minutes or 6 hours and 45 minutes to paint 75% of the pyramid.

According to the Venn diagram the value of [tex]n(A ^ C \cap B ^ C) = {3}[/tex] so the number of elements in that set is 1.

What is Venn diagram ?

A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. It is usually represented as a rectangle or a circle for each set and the overlapping areas between them, showing the common elements that belong to more than one set. Venn diagrams are widely used in mathematics, logic, statistics, and computer science to visualize the relationships between different sets and help solve problems related to set theory.

According to the question:
To solve this problem, we first need to understand the notation used.

n(A) denotes the set A and the numbers within the braces {} indicate the elements in set A. For example, n(A)={7,4,3,9} means that the set A contains 7, 4, 3, and 9.

n(AnB) denotes the intersection of sets A and B, i.e., the elements that are common to both A and B. For example, n(AnB)={4,3} means that the sets A and B have 4 and 3 in common.

^ denotes intersection of sets

cap denotes the intersection of sets

Now, we need to find the elements that are common to sets A and C, and sets B and C. We can do this by taking the intersection of A and C, and the intersection of B and C, and then taking the intersection of the two resulting sets.

[tex]n(A ^ C) = n(A \cap C) = {3,9}[/tex]

[tex]n(B ^ C) = n(B\cap C) = {3,5}[/tex]

Now, we take the intersection of [tex]n(A ^ C)[/tex] and [tex]n(B ^ C)[/tex]:

[tex]n(A ^ C \cap B ^ C) = {3}[/tex]

Therefore, the answer is 1.

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Answer:

$31,142

Step-by-step explanation:

To convert a percentage into a decimal, you move the decimal two places to the left. 1354% converted into a decimal is 13.54.

$23,000 * 13.54 = $31,142

to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below

[tex](\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{2}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{(-3)}}} \implies \cfrac{-6}{3 +3} \implies \cfrac{ -6 }{ 6 } \implies - 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{- 1}(x-\stackrel{x_1}{(-3)}) \implies y -2 = - 1 ( x +3) \\\\\\ y-2=-x-3\implies {\Large \begin{array}{llll} y=-x-1 \end{array}}[/tex]

Answer: Questionnaire refers to a research instrument, in which a series of question, is typed or printed along with the choice of answers, expected to be marked by the respondents, used for survey or statistical study. It consists of aformalisedd set of questions, in a definite order on a form, which are mailed to the respondents or manually delivered to them for answers. The respondents are supposed to read, comprehend and give their responses, in the space provided.

A ‘Pilot Study’ is advised to be conducted to test the questionnaire before using this method. A pilot survey is nothing but a preliminary study or say rehearsal to know the time, cost, efforts, reliability and so forth involved in it.

The interview is a data collection method wherein a direct, in-depth conversation between interviewer and respondent takes place. It is carried out with a purpose like a survey, research, and the like, where both the two parties participate in the one to one interaction. Under this method, oral-verbal stimuli are presented and replied by way of oral-verbal responses.

It is considered as one of the best methods for collecting data because it allows two way exchange of information, the interviewer gets to know about the respondent, and the respondent learns about the interviewer. There are two types of interview:

Personal Interview: A type of interview, wherein there is a face to face question-answer session between the interviewer and interviewee, is conducted.

Telephonic Interview: This method involves contacting the interviewee and asking questions to them on the telephone itself.

The critical values, the intervals of increasing or decreasing and the maximum and minimum points of the f(x) is (-1.5, -16), x < -1.5 and x = -1.5 and for b (4,6) and (2,10), (2,4).

A) Critical values

We will find out the critical value by solving for f ' (x) = 0

therefore, taking the derivative of given function we get,

f ' (x) = 4(2x) + 12 = 0

        = 8x + 12 = 0

therefore, 8x = -12

                 x = -12/8

                 x= -1.5

x = -1.5 is the only critical value in x-coordinate. Now to determine the y-coordinate, simply put the value of x in the function f(x) = 4x2 + 12x - 7

we get, f(-1.5) = 4(-1.5)2 + 12 (-1.5) - 7

                      = 4(2.25) - 18 - 7

                      = 9 - 25 = -16  

therefore, the critical value of the function f(x) = 4x2 + 12x - 7 is (-1.5, -16)

f(x) =x3 - 9x2 + 24x - 10.

Intervals of increasing and decreasing function is i.e. f decreases for

x < -1.5.

Therefore, f has minimum value at x = -1.5.

B) Critical values

We will find out the critical value by solving for f ' (x) = 0

therefore, taking the derivative of given function we get,

f '(x) = 3x2 - 9(2x) + 24

       = 3x2 - 18x + 24 = 0

therefore, 3 ( x2 - 6x + 8) = 0

   i.e x2 - 6x + 8 = 0

        (x-4) (x-2) = 0

So, x = 4 or x = 2 are the two critical values in x-coordinate. Now to determine the y-coordinate, simply put the values of x in the function f(x) =x3 - 9x2 + 24x - 10

we get, Substituting x = 4

f(4) = 43 - 9 (4)2 +24 (4) -10

     = 64 - 144 + 96 - 10

     = 6

Now, Substituting x = 2

f(2) = 23 - 9(2)2 + 24(2) - 10

     = 8 - 36 + 48 - 10

     = 10

Therefore, the critical values of the function f(x) =x3 - 9x2 + 24x - 10 are (4,6) and (2,10).

Intervals of increasing and decreasing functions is f decreases in (2,4).

therefore, f has minimum at x = 4 and maximum at x = 2.

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Complete question:

For each function determine: i) the critical values ii) the intervals of increasing or decreasing iii) the maximum and minimum points.

a. f(x) = 4x²+12x–7 (3 marks)

b. F(x) = x°-9x²+24x-10 (3 marks)

In response to the stated question, we may state that The margin of error function is equal to the highest mistake on the estimate.

In mathematics, a function is a connection between two sets of numbers in which each member of the first set (known as the domain) corresponds to a single element in the second set (called the range). In other words, a function takes inputs from one set and produces outputs from another. Inputs are commonly represented by the variable x, whereas outputs are represented by the variable y. A function can be described using an equation or a graph. The equation y = 2x + 1 represents a linear function in which each value of x yields a distinct value of y.v

(a) The population distribution must be assumed to be normal before generating a confidence interval.

(b) The margin of error with 90% confidence is provided by:

Error Margin = Z (/2) * (/n)

Where Z (/2) is the confidence level/2 crucial value, is the population standard deviation (unknown), and n is the sample size.

Error Margin = t (/2, n-1) * (s/n)

Where t (/2, n-1) is the critical value for the degrees of freedom /2 and n-1, and s is the sample standard deviation.

(d) The margin of error is equal to the highest mistake on the estimate.

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The value of the x is 5√3 after we successfully do the application of the 30°-60°-90° Triangle theorem.

The 30°-60°-90° Triangle Theorem states that in such a triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is the product of the length of the hypotenuse and the square root of 3 divided by 2.

Using this theorem, we can write:

y = hypotenuse

Opposite of 30° angle = 5 = hypotenuse/2

Opposite of 60° angle = x = hypotenuse × (√(3)/2)

Solving for the hypotenuse in terms of y from the first equation, we get:

hypotenuse = 5×2 = 10

Substituting this value into the third equation, we get:

x = 10 × (√(3)/2) = 5 × √(3)

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a.(1) When both r and s are odd integers, the quadratic polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers.

(2) When both r and s are even integers, the polynomial obtained by multiplying out (x - r)(x - s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers.

(3) When one of r and s is even and the other is odd, the polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, the coefficient of x term will be an odd integer, while the constant term will be an even integer.

b. x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.

a. When both r and s are odd integers, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers. This is because the sum of two odd integers and the product of two odd integers is also an odd integer.

When both r and s are even integers, the product (x − r)(x − s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers. This is because the sum of two even integers and the product of two even integers is also an even integer.

When one of r and s is even and the other is odd, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and the coefficient of x term will be an odd integer, while the constant term will be an even integer. This is because the sum of an odd and even integer is an odd integer, and the product of an odd and even integer is an even integer.

b. If x^2 - 1253x + 255 can be written as a product of two polynomials with integer coefficients, then we can write it as (x - r)(x - s) where r and s are integers. From part (a), we know that both r and s cannot be odd integers since the coefficient of x term would be odd, but 1253 is an odd integer. Similarly, both r and s cannot be even integers since the constant term would be even, but 255 is an odd integer. Therefore, one of r and s must be odd and the other must be even. However, the difference between an odd integer and an even integer is always odd, so the coefficient of x term in the product (x - r)(x - s) would be odd, which is not equal to the coefficient of x term in x^2 - 1253x + 255. Hence, x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.

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Answer:

height = 7 yards

Step-by-step explanation:

the area (A) of a trapezoid is calculated as

A = [tex]\frac{1}{2}[/tex] h (b₁ + b₂ )

where h is the perpendicular height between the 2 parallel bases

here b₁ = 12 , b₂ = 9 and A = 73.5 , then

[tex]\frac{1}{2}[/tex] h(12 + 9) = 73.5 ( multiply both sides by 2 to clear the fraction )

h(21) = 147

21h = 147 ( divide both sides by 21 )

h = 7 yards

Answer:

father age 45 son age 0 this is answer

The expression that is not equivalent to the model shown is given as follows:

-4(3 + 2). -> Option C.

Equivalent expressions are mathematical expressions that have the same value, even though they may look different. In other words, two expressions are equivalent if they produce the same output for any input value.

The expression for this problem is given by three times the subtraction of four, plus three times the addition of 2, hence:

3(-4) + 3(2) = -12 + 6 = 3(-4 + 2) = 3(-2) = -6.

Hence the expression that is not equivalent is the expression given in option C, for which the result is given as follows:

-4(3 + 2) = -4 x 5 = -20.

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Answer:

Show your solution ( 3. ) C + 18 = 29

Step-by-step explanation:

To solve the equation C + 18 = 29, we want to isolate the variable C on one side of the equation.

We can start by subtracting 18 from both sides of the equation:

C + 18 - 18 = 29 - 18

Simplifying the left side of the equation:

C = 29 - 18

C = 11

Therefore, the solution to the equation C + 18 = 29 is C = 11.

The required probability that a household in Maryland with annual income of ,

$90,000 or more is equal to 0.3377.

$50,000 or less is equal to 0.2218.

Annual household income in Maryland follows a normal distribution ,

Median =  $75,847

Standard deviation = $33,800

Probability of household in Maryland has an annual income of $90,000 or more.

Let X be the random variable representing the annual household income in Maryland.

Then,

find P(X ≥ $90,000).

Standardize the variable X using the formula,

Z = (X - μ) / σ

where μ is the mean (or median, in this case)

And σ is the standard deviation.

Substituting the given values, we get,

Z = (90,000 - 75,847) / 33,800

⇒ Z = 0.4187

Using a standard normal distribution table

greater than 0.4187  as 0.3377.

P(X ≥ $90,000)

= P(Z ≥ 0.4187)

= 0.3377

Probability that a household in Maryland has an annual income of $90,000 or more is 0.3377(rounded to four decimal places).

Probability that a household in Maryland has an annual income of $50,000 or less.

P(X ≤ $50,000).

Standardizing X, we get,

Z = (50,000 - 75,847) / 33,800

⇒ Z = -0.7674

Using a standard normal distribution table

Probability that a standard normal variable is less than -0.7674 as 0.2218. This implies,

P(X ≤ $50,000)

= P(Z ≤ -0.7674)

= 0.2218

Probability that a household in Maryland has an annual income of $50,000 or less is 0.2218.

Therefore, the probability with annual income of $90,000 or more and  $50,000 or less is equal to 0.3377 and 0.2218 respectively.

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The cardinality of set A, n(A) = 29

The cardinality of a set is the total number of elements in the set

Given the Venn diagram here shows the cardinality of each set. To find the cardinality of set A, n(A), we proceed as follows.

Since the cardinality of a set is the total number of elements in the set, then cardinality of set A , n(A) = 9 + 8 + 3 + 9

= 29

So, n(A) = 29

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Answer:

We can use the formula for the area of a rectangle to solve this problem. Let's assume that the length of the top of the bookcase is L and the width is b. Then, we can write:

L × b = 300

Solving for b, we get:

b = 300 / L

Since we don't know the length L, we cannot find the exact value of b. However, we can use the given information to make an estimate. Let's say that the length of the bookcase is 60 inches. Then, we have:

b = 300 / 60 = 5

So, if the length of the bookcase is 60 inches, the width needs to be at least 5 inches to accommodate Bria's soap carving collection. However, if the length is different, the required width will also be different.

Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.

Having three straight sides and three angles where they intersect, a triangle is a closed, two-dimensional shape. It is one of the fundamental geometric shapes and has a number of characteristics that can be used to study and resolve issues that pertain to it. The triangle inequality theory states that the sum of a triangle's interior angles is always 180 degrees, and that the longest side is always the side across from the largest angle. Triangles can be used to solve a wide range of mathematical issues in a variety of disciplines and can be categorised based on the length of their sides and the measurement of their angles.

given

We can use the following congruence theories or postulates based on the data in the diagram:

A. ASA

B. AAS

C. LL (corresponding angles hypothesis)

F. SAS

Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.

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The statement that best describes the mode of selection depicted in the figure is (b) Directional Selection, changing the average color of population over time.

The Directional selection is a type of natural selection that occurs when individuals with a certain trait or phenotype are more likely to survive and reproduce than individuals with other traits or phenotypes.

In the directional selection of evolution, the mean shifts that means average shifts to one extreme and supports one trait and leads to eventually removal of the other trait.

In this case, one end of the extreme-phenotypes which means that the dark-brown rats are being selected for. So, over the time, the average color of the rat population will change.

Therefore, the correct option is (b).

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The given question is incomplete, the complete question is

The figure below shows the change of a population over time. which statement best describes the mode of selection depicted in the figure?

(a) Disruptive Official, favoring the average individual

(b) Directional Selection, changing the average color of population over time

(c) Directional selection, favoring the average individual

(d) Stabilizing Selection, changing the average color of population over time

To highlight the line y = 0 on the graph in black/grey, draw a straight line passing through all points whose y-coordinate is 0.

What is graph?

In mathematics, a graph is a visual representation of a set of data, typically as a set of points or lines on a coordinate plane. Graphs are used to represent various types of data, such as numerical values, functions, relationships, and patterns.

Assuming that the graph is a coordinate plane with the x-axis and y-axis, do the following:

To highlight the point (9, 8) on the graph in red, locate the point (9, 8) on the coordinate plane and mark it with a red color.

To highlight the point (20, f(20)) on the graph in green, you need to know the value of f(20) first. Once you have that value, locate the point (20, f(20)) on the coordinate plane and mark it with a green color.

To highlight the line y = 5 on the graph in blue, draw a straight line passing through all points whose y-coordinate is 5. This line should be parallel to the x-axis and should be marked with a blue color.

Therefore To highlight the line y = 0 on the graph in black/grey, draw a straight line passing through all points whose y-coordinate is 0. This line should be parallel to the x-axis and should be marked with a black/grey color.

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a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. b. 20! = 20 x 19 x 18 x ... x 2 x 1. c. There are 16 zeros at the end of the decimal representation of (20!)2.

a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

b. 20! = 20 x 19 x 18 x ... x 2 x 1. To write this in factored form, we can identify the prime factors of each number and write the product using exponents. For example, 20 = 2² x 5, so we can write 20! as:

20! = (2² x 5) x 19 x (2 x 3²) x 17 x (2² x 7) x 13 x (2 x 2 x 3) x 11 x (2³) x (3) x (2) x 7 x (2) x 5 x (2) x 3 x 2 x 1

Simplifying, we get:

20! = 2¹⁸ x 3⁸ x 5⁴ x 7² x 11 x 13 x 17 x 19

c. The number of zeros at the end of (20!)² in decimal form is determined by the number of factors of 10, which is equivalent to the number of factors of 2 x 5. Since there are more factors of 2 than 5 in the prime factorization of (20!)², we only need to count the number of factors of 5. There are four factors of 5 in the prime factorization of 20!, which contribute four factors of 10 to the square. Therefore, (20!)²ends in 8 zeros when written in decimal form.

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The sample mean is 2.1, the sample variance is 212.5% and the standard deviation is 14.57%

a. The sample mean can be computed as the average of the quarterly percent total returns:

[tex](11.2 - 20.5 + 13.2 + 12.6 + 9.5 - 5.8 - 17.7 + 14.3) / 8 = 2.1[/tex]

So the sample mean is 2.1%, which can be interpreted as the average quarterly percent total return for the stock over the sample period.

b. The sample variance can be computed using the formula:

[tex]s^2 = sum((x - mean)^2) / (n - 1)[/tex]

where x is each quarterly percent total return, mean is the sample mean, and n is the sample size. Plugging in the values, we get:

[tex]s^2 = (11.2 - 2.1)^2 + (-20.5 - 2.1)^2 + (13.2 - 2.1)^2 + (12.6 - 2.1)^2 + (9.5 - 2.1)^2 + (-5.8 - 2.1)^2 + (-17.7 - 2.1)^2 + (14.3 - 2.1)^2 / (8 - 1) = 212.15[/tex]

So the sample variance is 212.15%. The sample standard deviation can be computed as the square root of the sample variance:

[tex]s = \sqrt(s^2) = \sqrt(212.15) = 14.57[/tex]

So the sample standard deviation is 14.57%.

c. To construct a 95% confidence interval for the population variance, we can use the chi-square distribution with degrees of freedom n - 1 = 7. The upper and lower bounds of the confidence interval can be found using the chi-square distribution table or calculator, as follows:

upper bound = (n - 1) * s^2 / chi-square(0.025, n - 1) = 306.05

lower bound = (n - 1) * s^2 / chi-square(0.975, n - 1) = 91.91

So the 95% confidence interval for the population variance is (91.91, 306.05).

d. To construct a 95% confidence interval for the standard deviation (as percent), we can use the formula:

lower bound = s * √((n - 1) / chi-square(0.975, n - 1))

upper bound = s * √((n - 1) / chi-square(0.025, n - 1))

Plugging in the values, we get:

lower bound = 6.4685%

upper bound = 20.1422%

So the 95% confidence interval for the standard deviation (as percent) is (6.4685%, 20.1422%).

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Answer:

Step-by-step explanation:

u got this

By using rational zeros theorem, we find that there are no real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, so we cannot factor f(x) over the real numbers.

To find the real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, we can use the rational zeros theorem, which states that any rational zeros of the function must have the form p/q, where p is a factor of the constant term (-35) and q is a factor of the leading coefficient (1).

The possible rational zeros of f are therefore ±1, ±5, ±7, ±35. We can then test each of these values using synthetic division or long division to see if they are zeros of the function. After testing all of the possible rational zeros, we find that none of them are actually zeros of the function.

Therefore, we can conclude that there are no real zeros of the function f(x) = x^3 - x^2 - 37x - 35.

However, we could factor it into linear and quadratic factors with complex coefficients using the complex zeros of f(x). But since the problem only asks for factoring over the real numbers, we can conclude that the factored form of f(x) is:

f(x) = x^3 - x^2 - 37x - 35 (cannot be factored over the real numbers)

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a) An automorphism of a group G is a bijective map φ:G→G that preserves the group structure. That is, φ(ab) = φ(a)φ(b) and φ(a⁻¹) = φ(a)⁻¹ for all a, b ∈ G.

b) The set-theoretic composition φψ of any two automorphisms φ, ψ is an automorphism, as it preserves the group structure and is bijective.

c) The set Aut(G) of all automorphisms of G, with the operation of composition of maps, is a group. This is because it satisfies the four group axioms: closure, associativity, identity, and inverses. Therefore, Aut(G) is a group under composition of maps.

An automorphism of a group G is a bijective map φ:G→G that preserves the group structure, meaning that for any elements a,b∈G, we have φ(ab) = φ(a)φ(b) and φ(a⁻¹) = φ(a)⁻¹. In other words, an automorphism is an isomorphism from G to itself.

To show that the set-theoretic composition φψ is an automorphism, we need to show that it satisfies the two conditions for being an automorphism. First, we have

(φψ)(ab) = φ(ψ(ab)) = φ(ψ(a)ψ(b)) = φ(ψ(a))φ(ψ(b)) = (φψ)(a)(φψ)(b)

using the fact that ψ and φ are automorphisms. Similarly,

(φψ)(a⁻¹) = φ(ψ(a⁻¹)) = φ(ψ(a))⁻¹ = (φψ)(a)⁻¹

using the fact that ψ and φ are automorphisms. Therefore, φψ is an automorphism.

To show that Aut(G) is a group, we need to show that it satisfies the four group axioms

Closure: If φ,ψ∈Aut(G), then φψ is also in Aut(G), as shown above.

Associativity: Composition of maps is associative, so (φψ)χ = φ(ψχ) for any automorphisms φ,ψ,χ of G.

Identity: The identity map id:G→G is an automorphism, since it clearly preserves the group structure and is bijective. It serves as the identity element in Aut(G), since φid = idφ = φ for any φ∈Aut(G).

Inverses: For any automorphism φ∈Aut(G), its inverse φ⁻¹ is also an automorphism, since it is bijective and preserves the group structure. Therefore, Aut(G) is closed under inverses.

Since Aut(G) satisfies all four group axioms, it is a group under composition of maps.

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If "f" is differentiable and f(1) < f(2), then there is a number "c", in the interval  (1, 2)  such that f'(c)>  0.

Applying the  Mean Value Theorem for derivatives, if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that:

f'(c) = (f(b) - f(a)) / (b - a)

In the scenario above, we have that f is differentiable, and that f(1) < f(2).

choosing a = 1 and b = 2.

Then applying the Mean Value Theorem, there exists at least one number c in the interval (1, 2) such that:

f'(c) = (f(2) - f(1)) / (2 - 1)

f'(c) = f(2) - f(1)

We have that f(1) < f(2), we have:

f(2) - f(1) > 0

We can conclude by saying that there exists a number c in the interval (1, 2) such that:

f'(c) = f(2) - f(1) > 0

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Answer:

See below.

Step-by-step explanation:

We can solve this linear programming problem using the simplex method. We will start by converting the problem into standard form

Maximize z = 3x₁ + 5x₂ + 0s₁ + 0s₂

subject to

x₁ - 5x₂ + s₁ = 35

3x₁ - 4x₂ + s₂ = 21

x₁, x₂, s₁, s₂ ≥ 0

Next, we create the initial tableau

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 -5 1 0 35

s₂ 3 -4 0 1 21

z -3 -5 0 0 0

We can see that the initial basic variables are s₁ and s₂. We will use the simplex method to find the optimal solution.

Step 1: Choose the most negative coefficient in the bottom row as the pivot element. In this case, it is -5 in the x₂ column.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 -5 1 0 35

s₂ 3 -4 0 1 21

z -3 -5 0 0 0

Step 2: Find the row in which the pivot element creates a positive quotient when each element in that row is divided by the pivot element. In this case, we need to find the minimum positive quotient of (35/5) and (21/4). The minimum is (21/4), so we use the second row as the pivot row.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 4/5 0 1/5 1 28/5

x₂ -3/4 1 0 -1/4 -21/4

z 39/4 0 15/4 3/4 105

Step 3: Use row operations to create zeros in the x₂ column.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 0 1/4 7/20 49/10

x₂ 0 1 3/16 -1/16 -21/16

z 0 0 39/4 21/4 525/4

The optimal solution is x₁ = 49/10, x₂ = 21/16, and z = 525/4.

Therefore, the maximum value of z is 525/4, which occurs when x₁ = 49/10 and x₂ = 21/16.

Answer:

[tex]14 {y}^{2} \sqrt{ {x}^{3} {z}^{9} }[/tex]

Step-by-step explanation:

[tex] \sqrt{196 {x}^{3} {y}^{4}{z}^{9} }= \sqrt{196} \times \sqrt{ {x}^{3} } \times \sqrt{ {y}^{4} } \times \sqrt{ {z}^{9} } \\ \sqrt{196} = 14 \\ \sqrt{ {x}^{3} } = {x}^{ \frac{3}{2} } \\ \sqrt{ {y}^{4} } = {y}^{2} \\ \sqrt{ {z}^{9}} = {z}^{ \frac{9}{2} }[/tex]

A fractional exponent is not necessarily simpler so just take out the 1st and 3rd parts of the term which simplify nicely:

[tex] \sqrt{196 {x}^{3} {y}^{4}{z}^{9} } = 14 {y}^{2} \sqrt{ {x}^{3} {z}^{9} } [/tex]

The scale factor of dilation of the shaded in gray to the shaded in black is 3

Given that

The preimage = shaded in gray

The image = shaded in black

From the graph, we have the following values on the image and the preimage

The preimage = shaded in gray = 4

The image = shaded in black = 12

The scale factor of dilation is then calculated as

Scale factor = shaded in black/shaded in gray

So, we have

Scale factor = 12/4

Evaluate

Scale factor = 3

Hence, the scale factor dilation is 3

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