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How does the function f(x)=x^2−4x−21 illustrate the Fundamental Theorem of Algebra?

Fill in the blanks. (Hint: Your answers are numbers.)

The degree of f(x) is _____. The Fundamental Theorem of Algebra guarantees that a polynomial equation has the same number of complex roots as its degree. This means that f(x) has exactly _____ zeros. Those zeros are 7 and _____.

The solution to the differential equation y' + 3y = f(t), y(0) = α using Laplace transforms is y(t) = αe⁻³ᵗ + F(s)/(s+3), where F(s) is the Laplace transform of f(t).

We use the Laplace transform on both sides of the problem in order to solve the given differential equation. Let Y(s) and F(s) represent the Laplace transforms of y(t) and f(t), respectively. Taking the Laplace transform of both sides of the equation, we have:

sY(s) - y(0) + 3Y(s) = F(s)

Substituting y(0) = α, we get,

sY(s) - α + 3Y(s) = F(s)

Rearranging the equation and solving for Y(s), we have,

Y(s) = (F(s) + α)/(s + 3)

Now, we need to find the inverse Laplace transform of Y(s) to obtain y(t). Using the properties of Laplace transforms, we know that the inverse Laplace transform of Y(s) is y(t) = L⁻¹{Y(s)}. Applying the inverse Laplace transform, we find,

y(t) = αe⁻³ᵗ + L⁻¹{F(s)/(s+3)}

The term αe⁻³ᵗ corresponds to the initial condition y(0) = α. The remaining term L⁻¹{F(s)/(s+3)} represents the inverse Laplace transform of F(s)/(s+3), which depends on the specific function f(t) and its Laplace transform.

Therefore, the solution to the differential equation is y(t) = αe⁻³ᵗ + F(s)/(s+3), where F(s) is the Laplace transform of f(t).

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Answer:

3 : 1

Step-by-step explanation:

Answer:

x = 10, x = 124

Step-by-step explanation:

(5)

Since Δ JKL is isosceles then the base angles are congruent, that is

∠ LJK = ∠ LJG = 80° , thus

∠ JLK = 180 - (80 + 80) = 180 - 160 = 20

x = 20 ÷ 2 = 10

(6)

Since Δ PQR is equilateral, then each of its angles = 60°

∠QRS = 60° - 32° = 28°

Since Δ SQR is isosceles then base angles are congruent, then

∠ SQR = ∠ QRS = 28 , thus

x = 180 - (28 + 28) = 180 - 56 = 124

answer :

x = 10, x = 124

-7 and - 9

Step-by-step explanation:

Area of rectangule is length * width

Answer:

39.4 or 40 % depending on which place you round it to

Step-by-step explanation:

18 * 1.10= 19.8      

(19.8  - 12) / 19.8 = 0.39393939...

so 39.4 or 40 % depending on which place you round it to

The illegal radio station from monitoring stations A and B is 33.49 miles and 27.03 miles respectively.      

As the graph triangle shows

∠CAB = 90° - 48° = 42°

∠CBA = 90° - 34° = 56°

So ∠ACB = 180° - ∠CAB - ∠CBA

               = 180° - 42° - 56°

               = 82°

AC/ sin ∠CBA = BC/ sin ∠CAB = AB/ sin ∠ACB

         AC = AB sin  ∠CBA / sin ∠ACB

              = 40 × sin 56°/ sin 82°

               = 33.49 miles

A mile is defined as the unit of length, which is exactly equal to 5280 feet, or 1760 yards, and standardized as exactly 1609.344 meters by the International agreement in 1959.

        BC = AB sin ∠CAB /sin ∠ACB

              = 40 × sin 42°/ sin 82°

              = 27.03 miles  

The illegal radio station from monitoring stations A and B is 33.49 miles and 27.03 miles respectively.

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The amount of honey each jar holds is 1/2 cup by jar

To solve this exercise we have to perform a mathematical operation of division with fractions:

Information about the problem:

To find the amount of honey each jar holds we state the following equation:

Amount of honey by jar = Cups brought / Jars filled

Amount of honey by jar = 14 cups /28 jar

Simplifying to its minimum expression, we have:

Amount of honey by jar = 7 cups / 14 jar

Amount of honey by jar = 1 cup / 2 jar

Amount of honey by jar = 1/2 cup by jar

Is a number that expresses the portion of some number over a total. The number that expresses the portion is known as the numerator and the number that expresses the total is known as the denominator.

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The conclusion drawn from the data best describes a positive correlation between height and weight for the team.

The table represents the heights and weights of the starting offensive players for a high school varsity football team. The question is asking for the conclusion that best describes the correlation between height and weight for the team.
                            To determine the correlation between height and weight, we can look at the data in the table and see if there is a pattern or trend. We can do this by creating a scatter plot of the data points, with height on the x-axis and weight on the y-axis.
                             After analyzing the scatter plot, we can draw the conclusion that there is a positive correlation between height and weight for the team. This means that as height increases, weight tends to increase as well. The data points on the scatter plot should show a general upward trend.

In summary, the conclusion drawn from the data best describes a positive correlation between height and weight for the team.

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The correlation between height and weight for the high school varsity football team can be described as positive, no correlation, or weak correlation based on the observations. Correlation only describes the relationship between the variables and does not imply causation or provide an explanation.

Based on the given table representing the heights and weights of the starting offensive players for a high school varsity football team, we can draw the following conclusion regarding the correlation between height and weight for the team:

1. Positive correlation: If we observe that as the heights of the players increase, their weights also tend to increase, then we can conclude that there is a positive correlation between height and weight. This means that taller players generally have higher weights, and vice versa.

2. No correlation: On the other hand, if we notice that there is no clear pattern or relationship between height and weight, with some tall players having low weights and vice versa, then we can conclude that there is no correlation between height and weight for the team.

3. Weak correlation: If there is a weak correlation between height and weight, it means that there is a slight tendency for taller players to have higher weights, but the relationship is not very strong or consistent. In this case, we might observe some tall players with lower weights and some shorter players with higher weights.

Correlation only describes the relationship between two variables, in this case, height and weight. It does not imply causation or explain why the correlation exists.

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Answer:

Step-by-step explanation:

Multiply both numbers by 4. That is your new coordinates. (12,-20)

Answer:

99 TVs

Step-by-step explanation:

5 and a half hours is about 330 minutes.

9:30 = x:330

If 30 * 11 = 330, then 9 * 11 will equal x.

Judy should inspect 99 TVs.

The values of x and y using the concept of similar figures are:

y = 166 and x = 3.33 units

Two triangles are said to be similar if their corresponding side proportions are the same and their corresponding pairs of angles are the same. When two or more figures have the same shape but different sizes, such objects are called similar figures.  

We are told that Polygon ABCD is similar to Polygon EFGH and as such applying the similar figure definition above, we can say that:

∠B = ∠F

Thus:

∠B = 360 - (61 + 116 + 90)

∠B = 360 - 267

∠B = 93°

Thus:

y - 73 = 93

y = 166

Using the concept of similar figures, then:

6/4 = 5/x

x = 20/6

x = 3.33 units

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Answer:

78 apples for 12 pies

Step-by-step explanation:

we use 13 apples for 2 pies

she makes 12 pies

6x2 = 12 pies

so we do 13x6

13x6=78

78 apples for 12 pies :)

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Answer:

13/2 = 6.5

12 x 6.5 = 78

? = 78

13/2 = 78/12

Step-by-step explanation:

13 apples ÷ 2 pies = 6.5 apples per pie

12 pies x 6.5 apples per pie = 78 apples for 12 pies

? = 78 apples for 12 pies

13/2 = 78/12

Those methods involving the collection, presentation, and characterization of a set of data in order to properly describe the various features of that set of data are called descriptive statistics.

Descriptive statistics:- Descriptive statistics helps to describe, summarize and organize the set of data efficiently and in an informative way, so that it'll be easier to make conclusion about the data in order to make rational decisions. This method describes the characteristics of a dataset.

Example:- The average test score for the students of a particular class, gives descriptive sense of the typical scores.

There are three types of descriptive statistics-

Thus we can conclude that, those methods involving the collection, presentation, and characterization of a set of data in order to properly describe the various features of that set of data are called descriptive statistics.

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In Problem 4, there are (i) 495 different possible orders for the group when they collectively order four different dishes to share, and (ii) 20,736 different possible orders for the group when each individual orders a main course.

(i) To find the number of ways to order four different dishes out of 12, we use combinations. This is calculated as C(12,4) = 12! / (4! * (12-4)!), which equals 495 possible orders.

(ii) Since there are 12 dishes and each of the four friends can choose any dish, we use permutations. The number of possible orders is 12⁴, which equals 20,736 different orders.

For passwords, there are (i) 306,380,448 passwords of length 7 with at least one digit, and (ii) 282,475,249 passwords of length 7 with at least one digit and one letter.

(i) There are 10 digits and 26 lowercase letters. Total possibilities are (10+26)⁷. Subtract the number of all-letter passwords: 26^7. Result is (36⁷) - (26⁷) = 306,380,448.

(ii) Subtract the number of all-digit passwords from the previous result: 306,380,448 - (10⁷) = 282,475,249 different passwords.

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the base-ten representation for 19! = 121,6T5,100,40M,832,000 then

H+M+T =12

19! has three 5s so it will end with three 0s.

Hence H = 0

19! = 121,6T5,100,40M,832,000 = 121,6T5,100,40M,832 * 1000

19! has many 2s. So M832 will be divisible by 16 (last 4 digits).

So M000 + 832 is divisible by 16. 832 is completely divisible by 16. Since 16*5 = 80, 8000 must be divisible by 16.

M = 8

19! = 121,6T5,100,408,832,000

Since 19! is divisible by 9, sum of all digits must be divisible by 9.

1+2+1+6+T+5+1+4+8+8+3+2 = T + 41

To make this divisible by 9, T must be 4.

H+M+T = 0 + 8 + 4 = 12

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Answer:

A. x^2 + y^2 = 4^2

Step-by-step explanation:

The equations of a circle is x^2 + y^2 = r^2

Therefore, the equation of this circle is x^2 +y^2 = 4^2

Answer:

A

Step-by-step explanation:

Remark

The formula for a circle is

x^2 + y^2 = r^2

The number associated with x^2 and y^2 must always be 1. If it's not then you do not have a circle.

Answer

The answer to this question is x^2 + y^2 = 4^2 where r = 4 and it is squared.

Real-world problem on additive inverse of a rational number and solution:

Peter borrowed one fourth of his salary to Jack. The money way returned to peter at the end of the month without interest because they are friends. how much is Jack owing peter in the transaction?

Jack is owing Peter nothing

Additive inverse is the opposite of a number in terms of addition, this is the number that gives zero when added to the original number.

To get a result equal to 0, add the original number and the additive inverse or by simply changing the sign of the integer. On the basis of negation of the original number.

In the problem let peter's salary be x

amount peter borrowed out = x/4

amount paid back = x/4

Addition of the expenses

borrowed + paid back

-x/4 + x/4 = 0

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The 84th derivative of 2sin(x) at x = -π/3 is -2cos(-π/3) = -2cos(π/3) = -2(-1/2) = 1.

The 84th derivative of 2sin(x) at the point x=-π/3 can be found using the chain rule and the fact that the derivative of sin(x) is cos(x). The chain rule states that the derivative of a function f(g(x)) is f'(g(x))g'(x). In this case, f(x) = 2sin(x) and g(x) = x, so f'(x) = 2cos(x) and g'(x) = 1. Therefore, the 84th derivative of 2sin(x) is 2cos(x) * 1 = 2cos(x).

To find the value of the 84th derivative at the point x=-π/3, we simply plug in -π/3 for x in the expression 2cos(x):

2cos(-π/3) = 2 * (1/2) = 1

Therefore, the 84th derivative of 2sin(x) at the point x=-π/3 is 1.

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Answer:

x=12

Step-by-step explanation:

10x-18+6x+6=180

16x-12=180

16x=180+12

16x=192

x=192/16

x=12

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The discriminant at (x, y) = (0, 1), (0, 2), (4, 1), and (1, 0) are 12, 32, 40, and 16, respectively.

The discriminant of a function f(x, y) is given by:

D = f{zx}f(x, y) - f{xy}²

where f{zx} and f{xy} are the partial derivatives of the function with respect to x and y, respectively.

The function f(x, y) = 2x² + 2xy + y² - 2x - 2y + 5, so we have:

f_{x} = 4x + 2y - 2

f_{y} = 2x + 2y - 2

f_{zx} = 4

f _{xy} = 2

At (x, y) = (0, 1), we have:

f(0, 1) = 4

f_{x}(0, 1) = -2

f_{y}(0, 1) = 0

f _ {zx}(0, 1) = 4

f_{xy}(0, 1) = 2

Therefore, the discriminant at (x, y) = (0, 1) is:

D = f{zx}f(x, y) - f{xy}²

D = 4 × 4 - 2²

D = 16 - 4

D = 12

Similarly, at (x, y) = (0, 2), we have:

f(0, 2) = 9

f_{x}(0, 2) = -2

f_{y}(0, 2) = 2

f_{zx}(0, 2) = 4

f_{xy}(0, 2) = 2

Therefore, the discriminant at (x, y) = (0, 2) is:

D = f{zx}f(x, y) - f{xy}²

D = 4×9 - 2²

D = 36 - 4

D = 32

Similarly, at (x, y) = (4, 1), we have:

f(4, 1) = 11

f_{x}(4, 1) = 14

f_{y}(4, 1) = 6

f_{zx}(4, 1) = 4

f_{xy}(4, 1) = 2

Therefore, the discriminant at (x, y) = (4, 1) is:

D = f{zx}f(x, y) - f{xy}²

D = 4×11 - 2²

D = 44 - 4

D = 40

Finally, at (x, y) = (1, 0), we have:

f(1, 0) = 5

f_{x}(1, 0) = 4

f_{y}(1, 0) = 2

f_{zx}(1, 0) = 4

f_{xy}(1, 0) = 2

Therefore, the discriminant at (x, y) = (1, 0) is:

D = f{zx}f(x, y) - f{xy}²

D = 4×5 - 2²

D = 20 - 4

D = 16

Hence, the discriminant at (x, y) = (0, 1), (0, 2), (4, 1), and (1, 0) are 12, 32, 40, and 16, respectively.

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Answer:

d

Step-by-step explanation:

Answer:

when x=2

-2×2+4y=16

4y=16+4

y=20/4=5

again

x=-8

-2×-8+4y=16

4y=16-16

y=0/4=0

Answer:

y = x^2 +2

Step-by-step explanation:

x^2 is parabolic not linear

To prove that in a collection of 51 integers there is a subset of 11 where the difference of two of them is a multiple of 5, we can use the Pigeonhole Principle.

Consider the residues of the integers modulo 5, which can take values from 0 to 4. Let's assume that none of the 51 integers has a difference that is a multiple of 5. In other words, no pair of integers in the collection has a difference that leaves a residue of 0 when divided by 5.

Now, we can divide the 51 integers into five subsets based on their residues modulo 5. Each subset will contain the integers with the corresponding residues 0, 1, 2, 3, or 4.

Since we have 51 integers and only five possible residues, by the Pigeonhole Principle, at least one of the subsets must contain at least 51 / 5 = 10 + 1 = 11 integers. Let's assume, without loss of generality, that the subset with residue 0 contains 11 or more integers.

Within this subset, any pair of integers will have a difference that leaves a residue of 0 when divided by 5. Hence, we have found a subset of 11 integers where the difference of two of them is a multiple of 5.

Therefore, we have proved that in a collection of 51 integers, there is always a subset of 11 where the difference of two of them is a multiple of 5.

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At nearly 15 percent, the proportion of foreign born U.S. residents is the highest it has been since 1910.

The statement suggests that the proportion of foreign-born U.S. residents is the highest it has been since 1910. This means that out of all the different groups mentioned (Southeast Asian, foreign-born, undocumented immigrant, American Indian), the proportion of foreign-born residents has reached the highest level.

"Foreign-born" refers to individuals who were born outside the United States and may have immigrated to the country. The proportion of foreign-born residents is a measure of the percentage of people in the United States who were born in another country.

The statement implies that the proportion of foreign-born residents, at 15 percent, is currently the highest it has been since 1910. This indicates that the United States is currently experiencing a higher level of immigration or an increased number of people born outside the country who have become residents.

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The question is incomplete the complete question is :

At 15 percent, the proportion of __________ U.S. residents is the highest it has been since 1910.

Southeast Asian

foreign born

undocumented immigrant

American Indian

The probability of choosing a blue or yellow highlighter when a yellow highlighter is picked is 17/25.

Here, according to the question:

Total number of yellow highlighters  = 12

Total number of blue  highlighters  = 6

Total number of green highlighters  = 4

Total number of orange  highlighters  = 3

∴ Total number of highlighters  =  12 + 6 + 4 + 3  = 25

Let E1: Event of selecting yellow highlighter.

P(E1) = \(\frac{Total number of yellow highlighters}{Total highlighters}\) = \(\frac{12}{25}\)

Let E2: Event of selecting blue or yellow highlighter one yellow highlighter is selected.

∴Total yellow highlighters left after selecting one = 12 -1  = 11

Also, the number of blue highlighters = 6

So, the TOTAL FAVORABLE to E2 = 6+ 11 = 17

P(E2)  = \(\frac{Total number of yellow + blue highlighters}{Total highlighters}\) = \(\frac{17}{25}\)

Hence, The probability of choosing a blue or yellow highlighter when a yellow highlighter is picked is 17/25.

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The sun would more frequently be reaching the ground at a high angle at latitude 0° (the equator). Option A is correct.

What is latitude?

Latitude is the angle north or south of the equator and extends horizontally over the globe. Longitude, on the other hand, runs from east to east and runs vertically across the globe.

On any given day, only points on Earth's surface that are located along a single line of latitude can experience sunlight at a 90-degree angle. Lower levels of sunshine are present everywhere else. The sun's beams are typically strongest at the equator and weakest at the poles.

Option A is correct,0° (the equator).

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Answer: That would be alternate interior angles

Step-by-step explanation:

The percentage of students passing the partial exam is:

Option c. 70%

To answer this question, we use the Z-table. We have:

Given, mean (μ) = 120 and standard deviation (σ) = 17
Passing Score = 111

To find out what percentage of students passed the midterm, we need to find the probability that a score picked at random from the distribution is greater than or equal to 111. We can convert this score into a standard score (Z-score) using the Z-table. We have:

Z-score = (111 - μ) / σ
       = (111 - 120) / 17
       = -0.53

Using the Z-table, we can find the area (probability) to the left of this Z-score as follows:

Area to the left of Z-score = 0.2981 (from Z-table)

The area to the right of this Z-score (i.e. the probability that a score picked at random from the distribution is greater than or equal to 111) is:

Area to the right of Z-score = 1 - 0.2981
                           = 0.7019
                           = 70.19%

Therefore, the percentage of students who passed the midterm is approximately 70% (Option C).

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Answer:

C. 70%

Step-by-step explanation: