Which binomial is a factor of f/x )= x³ 6x² +3x 10?

The volume of the rectangular prism can be calculated using the formula , l x b x h , where l = length , b = base and h is the height.

The SI unit for rectangular prism volume is m³.

A polyhedron which has two congruent and a parallel bases  is known as a  rectangular prism. It also known as a cuboid.

A rectangular prism has 6 faces and 12 edges. It is referred to as a prism because of the length of its cross-section.

We study prisms under the domain of geometry .

They are of two different types,

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The x-intercept is the point where a line crosses the x-axis, and the y-intercept is the point where a line crosses the y-axis.

Given data :

Find x-intercept

The x-intercept is where the graph crosses the x-axis. To find the x-intercept, we set y1=0 and then solve for x.

-4x + y = 16

-4x + 1(0) = 16

x1 = -4    y1 = 0

Find y-intercept

The y-intercept is where the graph crosses the y-axis. To find the y-intercept, we set x2=0 and then solve for y.

-4x + y = 16

-4(0) + y = 16

y2 = 16    x2 = 0

Solve for x

To solve for x, we solve the equation so the variable x is by itself on the left side:

-4x + y = 16

x = -4 - 0.25y

Solve for y

To solve for y, we solve the equation so the variable y is by itself on the left side:

-4x + y = 16

y = 16 + 4x

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The value of the sine of angle D to the nearest hundredth will be 0.25.

The Law of Sines The law of trigonometric functions, sine law, sine formula, or sinusoidal rule is a trigonometric equation that relates the sizes of any triangle's edges to the sines of its orientations.

The sine law in the triangle ΔCDE is given as,

CD / sin E = DE / sin C = EC / sin D

In triangle ΔCED, the measure of angle ∠E = 90°, ED = 63, DC = 65, and CE = 16. Then we have

65 / sin 90° = 63 / sin C = 16 / sinD

Compare the first and the last term, then we have

65 / sin 90° = 16 / sin D

65 = 16 / sin D

sin D = 16 / 65

D = 14.25°

The worth of the sine of point D to the closest 100th will be 0.25.

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The graph of 4x7 40x6 100x5 is a straight line that decreases in slope as x increases.

The graph of 4x7 40x6 100x5 is a linear graph with a positive slope. The graph starts from the point (4,7) and increases in a linear fashion. At the point (40,6) the value of y is 6 and the value of x is 40. The graph then continues to increase as the value of x increases. At the point (100,5) the value of y is 5 and the value of x is 100. The graph of 4x7 40x6 100x5 is a linear graph with a positive slope, increasing from 4x7 to 100x5, and having a y-intercept of 0.The graph then continues to decrease in slope as x increases and reaches the point (40,6) and finally (100,5). This graph is a linear equation which can be written as

y = mx + b

where m is the slope and b is the y-intercept. . The y-intercept is the point where the graph crosses the y-axis and can be found by substituting 0 for x in the equation.The slope is calculated by subtracting the y-values and dividing by the x-values, which gives us a slope of -1/96. The y-intercept is calculated by substituting 0 for x in the equation, which gives us a y-intercept of 7.

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{y | y ∈ ℝ} or (-∞, ∞) is the range of y = 3x + 1

To determine the range of a group of observations, firstly arrange them in ascending order. The range may then be determined by comparing the difference between the highest and least numbers.

A function's range includes all conceivable values for y.  y = f(x) is the formula to determine a function's range (x). It is only a function in a relationship if each x value has exactly one y value.

Think about the function y = f(x). The function's range is defined as the range of all y values, from least to maximum. Substitute all possible values of x into the provided expression of y to see if it is positive, negative, or equal to other values.

For, y = 3x + 1

domain is all real numbers

so, {x | x ∈ ℝ} or (-∞, ∞)

range:

y = 3x + 1

or, -3x = -y + 1

or, 3x = y - 1

or, x = (y - 1)/3

so, {y | y ∈ ℝ} or (-∞, ∞)

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The complete question is as follows:

What is the range of y = 3x + 1

Answer:

slope of line w = - [tex]\frac{8}{4}[/tex]

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = [tex]\frac{5}{8}[/tex] x + [tex]\frac{9}{4}[/tex] ← is in slope- intercept form

with slope m = [tex]\frac{5}{8}[/tex]

given a line with slope m then the slope of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{5}{8} }[/tex] = - [tex]\frac{8}{5}[/tex]

The coefficient of x^11 in the binomial expansion of (x - 3)^12 is -36.

What is binomial expansion?

The binomial theorem in basic algebra explains how a binomial's powers can be expanded algebraically.

Exponents of x and y added together always equal n. The binomial coefficients that are equally spaced from the start and end are equal, i.e., nC0 = nCn, nC1 = nCn-1, nC2 = nCn-2, etc.

Consider, the given expression (x - 3)^12

We have to find the coefficient of x^11 in the expansion of (x - 3)^12.

Applying the binomial theorem,

(a + b)^n =  Σ(i = 0 to n) nCi a^(n- i)b^i

Here a = x, b = -3.

⇒ (x - 3)^12 =  Σ(i = 0 to 12) 12Ci x^(12 - i)(-3)^i

[tex](x -3)^1^2 = x^ 1^2-36^1^1+59x^1^0 - 5940x^9 + 40095x^8 - 192456x^7+ 673596x^6 - 1732104x^5 + 3267695x^4-4330260x^3 + 3897234x^2- 2125764x+531441[/tex]

Hence, the coefficient of x^11 in the binomial expansion of (x - 3)^12 is -36.

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Utilize Excel's Advanced Filter to locate singular values. You have the option of extracting unique values or filtering for them. Use the magical UNIQUE function if you have Excel 365 or Excel 2021.

To develop a distinctive value proposition that complements a company's objectives and core principles, think about doing the following seven steps: 1. Examine the offerings of the business. When developing a distinctive value proposition, a company's products or services might serve as a source of inspiration. These services demonstrate the value the business provides to its clients.

Select a data set cell by clicking it.

Then, choose Advanced from the Sort&Filter group by clicking the Data tab.

Select "Copy to Another Location" from the menu.

Excel will show the List Range as the cell reference for the whole data set or Table object.

If there is a criteria range, remove it.

You should always provide a text label at the top of each data column when using Excel's Advanced Filter.

1. Select a list range cell by clicking it.

2. Click Advanced under the Sort & Filter category on the Data tab.

The dialog box for the Advanced Filter opens.

3. Click the Copy button (see image below).

4. Select cell C1 by clicking in the Copy to box.

5. Verify only unique records.

6. Press OK.

Lion in cell A7 and Elephant in cell A9 are two examples of duplicate values that Excel eliminates before sending

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Answer:

C. All real number

Step-by-step explanation:

Most of the f(x) functions have the domain of (negative infinity, positive infinity) and the function in the picture is the same. So, the answer is C

The nature of the roots of the given equation is real, distinct and rational.

The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.

Given that a quadratic equation, 3x²+2x-1 = 0,

To find the nature of the roots, we will find the discriminant, of the equation,

We know that,

1) A positive discriminant indicates that the quadratic has two distinct real number solutions.

2) A discriminant of zero indicates that the quadratic has a repeated real number solution.

3) A negative discriminant indicates that neither of the solutions are real numbers.

D = √b²-4ac

b = 2, a = 3 and c = -1

D = √(2)²-4 × (-1) × (3)

D = √16

D = 4

Since, D > 0

Hence, the nature of the roots of the given equation is real, distinct and rational.

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Here the correct expression for the derivative of P(x) is: option B

The fact that the initial function f implies a second, distinct function f′(x) gives origin to the name "derivative" (x). As a result, we have deduced one from another. When we deduce one from the other, the terms differential, etc. pertain more to the real mathematics that is taking place.

The definition of a derivative in mathematics is a technique for displaying the simultaneous rate of change. The amount by which the provided function is changing at a certain instant is thus represented by it.

The following are some uses for derivatives: to ascertain the pace at which one item is changing in relation to another. The saddle points of a function are used to calculate maximum and minimum. to establish a function's concavity and convexity.

P(x) = [tex]\frac{f(x)}{1-f(x)}[/tex]

or, [tex]\frac{d}{dx} P(x) =[/tex] [tex]\frac{d}{dx} \frac{f(x)}{1-f(x)}[/tex]

or, [tex]\frac{d}{dx} P(x) =[/tex] [tex]\frac{f'(x)}{1-f(x)}[/tex]

Correct option: B

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The algebraic expression for each student is 88/x

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

Number of baskets = 8

Pears in each basket = 11

The total number of pear is calculated as

The total number of pear = Number of baskets * Pears in each basket

Substitute the known values in the above equation, so, we have the following representation

The total number of pear = 8 * 11

Evaluate

The total number of pear = 88

Represent the students with x

So, the expression is 88/x

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The given statement w.r.t clinical trials is true.

Clinical trials are prospective biomedical or behavioral research studies on human participants designed to answer specific questions about biomedical or behavioral interventions, including new treatments and known interventions that warrant further study and comparison.

Given is a statement about the clinical trials.

The given statement that Clinical trials enroll individual subjects and enable randomization of subjects to receive or not receive the intervention is true.

Therefore, the given statement w.r.t clinical trials is true.

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The domain and scope of the identity function are the same. Equation for the identity function is  f(x) = x, or y = x.

An statement, rule, or law in mathematics that specifies the connection between an independent variable and a dependent variable (the dependent variable). A relationship between a group of inputs and one output each is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. Y=X2 is an illustration of this. You only receive one output for y if you enter anything for x. The fact that x is the input value leads us to argue that y is a function of x.

identity function are the same.

Equation for the identity function is

f(x) = x, or y = x.

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We can find the missing value of a triangle on a calculator in the following way,

If the value is a side of a triangle and the triangle is a right-angle triangle, we apply the Pythagoras Theorem. The formula of this theorem is a²+ b²= h², where h is the hypotenuse of the triangle, and a and b are the remaining two sides.

If the triangle is any other triangle, we use the law of sine, which is:

a/sin(A)= b/sin(B)= c/sin(C), where a, b, and c are sides of the triangle and A, B, and C are the angles opposite to the respective sides.

If the value is an angle of the triangle, we use the formula

x=180-(y+z), where x is the missing angle, and y and z are the remaining two angles of the triangle.

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Answer: i think b

Step-by-step explanation:

Answer: -1

Step-by-step explanation:

slope form is [tex]m = \frac{y^{2} - y^{1} }{x^{2} - x^{1} }.[/tex] M is the slope. x1 and y1 are coordinates of a point along the line. x2 and y2 are the coordinates of another point along the line.

So we need to substitute in the points you have. let's put in (1, -2) for x1 and y1, and (2, -3) in x2 and y2. it makes [tex]m = \frac{-3-(-2)}{2-1}[/tex]

Now all you have to do is solve the equation.

[tex]m = \frac{-3-(-2)}{2-1}[/tex]

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

m = -1

(btw the line's equation would be y = -x - 1)

Answer:

-1

Step-by-step explanation:

To find the slope between two points we can use the slope formula

m = ( y2-y1)/(x2-x1)

   = ( -2 - -3)/(1 -2)

   = ( -2 +3)/( 1-2)

  = 1/-1

  = -1

It will take 2 year the population to reach 300 students.

What is rounding to the nearest tenth?

Look at the hundredth number to round the decimal number to the nearest tenth. If it's more than 5, add one to the tenth value. Remove all the numbers that are present after the tenth place and leave the tenth place value alone if it is less than 5. To the nearest tenth, 4.6, for instance, is 4.624.

Given that, the initial number of students is 242.

The student population is growing at 12% a year.

The number of students that increases is 300 - 242 = 58.

Applying formula

I = PRT

I = The number of students that increased = 58

P = initial number of students = 242

T = Time

R = rate of increasing =12% = 0.12

Therefore,

58 = 242 × T × 0.12

T = 58/(242 × 0.12)

T = 1.997

T = 2.0 years

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The standard deviation of

{1,2,3,4,5}=[52−112]12

=√2

Standard Deviation is a measure which shows how much variation (such as spread, dispersion, spread,) from the mean exists.

The standard deviation indicates a “typical” deviation from the mean. It is a popular measure of variability because it returns to the original units of measure of the data set.

Like the variance, if the data points are close to the mean, there is a small variation whereas the data points are highly spread out from the mean, then it has a high variance.

Standard deviation calculates the extent to which the values differ from the average. Standard Deviation, the most widely used measure of dispersion, is based on all values.

Therefore a change in even one value affects the value of standard deviation. It is independent of origin but not of scale.

It is also useful in certain advanced statistical problems.

Let's develop a general formula then as a particular you get standard deviation of

1,2,3,4 and 5

If we have

{1,2,3,....,n}

and we need to find the standard deviation of this numbers.

Note that

Var(X)=1nn∑i=1 x2i−(1nn∑i=1xi)2

⇒Var(X)=1nn∑i=1i2−(1nn∑i=1 i)2

⇒Var(X)=1n⋅n(n+1)(2n+1)6−(1n⋅n(n+1)2)2

⇒Var(X)=(n+1)(2+1)6−(n+12)2

⇒Var(X)=n+12[2n+13−n+12]

⇒Var(X)=n+12⋅n−16

⇒Var(X)=n2−112

So, Standard deviation of

{1,2,3,....,n}

is

[Var(X)]12=[n2−112]12

In particular, your case the standard deviation of

{1,2,3,4,5}

=[52−112]12=√2.

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Coefficient of Standard Deviation vs. Coefficient of Variation: When to Use Each

The standard deviation is most commonly used when we want to know the spread of values in a single dataset.

However, the coefficient of variation is more commonly used when we want to compare the variation between two datasets.

For example, in finance the coefficient of variation is used to compare the mean expected return of an investment relative to the expected standard deviation of the investment.

For example, suppose an investor is considering investing in the following two mutual funds:

Mutual Fund A: mean = 9%, standard deviation = 12.4%

Mutual Fund B: mean = 5%, standard deviation = 8.2%

The investor can calculate the coefficient of variation for each fund:

CV for Mutual Fund A = 12.4% / 9% = 1.38

CV for Mutual Fund B = 8.2% / 5% = 1.64

Since Mutual Fund A has a lower coefficient of variation, it offers a better mean return relative to the standard deviation.

Here’s a brief summary of the main points in this article:

Both the standard deviation and the coefficient of variation measure the spread of values in a dataset.

The standard deviation measures how far the average value lies from the mean.

The coefficient of variation measures the ratio of the standard deviation to the mean.

The standard deviation is used more often when we want to measure the spread of values in a single dataset.

The coefficient of variation is used more often when we want to compare the variation between two different datasets.

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Answer:

4 nautical miles per hour

Step-by-step explanation:

Speed = Distance / Time

Distance = Speed x time

D1 = D2

S1 = 16 nautical miles per hour

t1  =10 hours

We know S2 is unknown and the t2 = 8 hours

So, we take

S1 x T1 = S2 x T2

16 x 10 = S2 x 8

S2 = 16 x 10 / 8

S2 = 20 nautical miles per hour

20 - 16 = 4 nautical miles per hour

So, it has to increase its speed by 4 nautical miles per hour

Answer: 4 nautical miles per hour

It is given that in the first case, the ship takes 10 hours to travel at the speed of 16 nautical miles per hour. distance=speed × time=16×10=160 nautical miles. Therefore, the answer to the above question is 4 nautical miles per hour.

Step-by-step explanation:

By completing squares, we can rewrite the quadratic equation as:

3*(x - 2)^2 = 4

Here we have the quadratic equation:

3x^2 - 12x + 8 = 0

If we take "3" as a common factor for the first two terms:

3*(x^2 - 4x) + 8 = 0

Now we can add and subtract 12, then we will get:

3*(x^2 - 4x) + 8 = 0

3*(x^2 - 4x) + 8  +12 - 12 = 0

We can put the positive one inside the parenthesis:

3*(x^2 - 4x + 4) + 8 - 12 = 0

3*(x - 2)^2 - 4= 0

3*(x - 2)^2 = 4

That is our quadratic equation, the correct option is the third one.

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Answer:

[tex]120b^2[/tex]

Step-by-step explanation:

When multiplying two terms, multiply the coefficients and the variables.

Also, remember the rules for multiplying integers:

-8 * -15 = +120

b * b = [tex]b^2[/tex]

∴(-8b)(-15b) = [tex]120b^2[/tex]

[tex](-8b)(-15b)[/tex]

[tex](-8)\times b\times(-15)\times b[/tex]

[tex]120\times b^2[/tex]

[tex]=120b^2[/tex]

[tex]\fbox{Option A}[/tex]

An angle is a figure found by two rays with the same initial point. In geometry, various names are given to different angles and their combinations depending upon the type of angles they make.

various geometrical shapes such as a triangle or a quadrilateral contain a set of angles that are governed by a set of rules.

For example, there are three interior angles in a triangle, the sum of which is always equal to 180 degrees. This means that if we are cultured of two of the angles in a triangle, we can find the third.

alike using different rules and the available information we can find the missing angle in a geometrical figure.

But, before understanding different state where we can find the missing angles, we should first recall some of the properties of different geometric shapes that will be useful in finding missing angles.

1. The sum of the measure of the three interior angles of a triangle is always 180 degrees.

2. If one of the angles of a triangle is 90 degrees, the sides that make the right angle is called the base and the perpendicular while the third side is called the hypotenuse.

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To multiply radical expressions with the same index, we use the product rule for radicals. [tex]\sqrt[n]{A}.\sqrt[n]{B} = \sqrt[n]{A.B}[/tex]

To divide radical expressions with the same index, we use the quotient rule for radicals. [tex]\frac{\sqrt[n]{A} }{\sqrt[n]{B} } =\sqrt[n]{\frac{A}{B} }[/tex]

Multiplying Radical Expressions :

Example,

given:  Multiply: [tex]\sqrt[3]{12} .\sqrt[3]{6}[/tex]

Apply the product rule for radicals, and then simplify.

[tex]\sqrt[3]{12}.\sqrt[3]{6}=\sqrt[3]{12.6}[/tex]

            [tex]=\sqrt[3]{72}\\=\sqrt[3]{2^{3} .3^{2} } \\=2\sqrt[3]{3^{2} } \\=2\sqrt[3]{9}[/tex]

Dividing Radical Expressions

Example,

given: Divide: [tex]\frac{\sqrt[3]{96} }{\sqrt[3]{6} }[/tex]

In this case, we can see that 6 and 96 have common factors. If we apply the quotient rule for radicals and write it as a single cube root, we will be able to reduce the fractional radicand.

[tex]\frac{\sqrt[3]{96} }{\sqrt[3]{6} } =\sqrt[3]{\frac{96}{6} }[/tex]

      [tex]=\sqrt[3]{16} \\=\sqrt[3]{8.2} \\=2\sqrt[3]{2}[/tex]

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The order of the volumes of the given figures from least to greatest is:

(iii) < (i) < (ii)

The first figure is a cone with a diameter = of 10 cm

radius = 10/2 = 5 cm

height = 30 cm

The second figure is a cone with the radius of = 20 cm

height = 10 cm

The third figure is a cylinder with the diameter = 10 cm

radius = 10/2 = 5 cm

height = 5 cm

We know that,

The volume of a cone = (1/3) * π * r² * h

The volume of a cylinder =  π * r² * h

Now, the volume of first cone = (1/3) * π * 5² * 30 cm³  =   250π cm³

Now, the volume of second cone = (1/3) * π * 20² * 10 cm³  = 1333π cm³

Now, the volume of cylinder = π * 5² * 5 = 125π cm³

So, the correct answer of volume from least to greatest is :

(iii) < (i) < (ii)

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Two ratios are referred to as being in proportion if they are the same. If the four elements are a, b, c, and d in that order, then a : b :: c : d = a/b = c/d.

The following formulas also relate to proportion:

a : b :: c : d = a/b = c/d

In which,

The ratio of averages to extremes is 1:1. You can write this as ad = bc.

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The line has a 4/5 slope. The solution is 4/5 if the line's equation is y = 4x/5 - 3.

The ratio in which y increases as x increases is known as a line's slope. The slope of a line not only shows how steep it is but also how much y increases as x increases. Along the entire line, there is a constant slope (the same).

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

We have a line equation.

y = (4/5)x - 3

or

y = 4x/5 - 3

A straight line, as we all know, is made up of countless points that are connected on both sides of the point.

The straight line's typical form is as follows:

y = mx + c

The slope of the line is m.

C is the line's y-intercept.

m = 4/5

c=-3

Consequently, the line's slope is 4/5. The solution is 4/5 if the line's equation is y = 4x/5 - 3.

The complete question is:-

What is the slope of the line represented by the equation y=4/5x-3?

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The slope of the line can be calculated as  [tex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/tex].

Correct options are (A) and (D).

What is the slope of the line?

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

As the definition of the slope says that it is the ratio of rise by run.

so we can make the formula as

[tex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

Hence, the slope of the line can be calculated as  [tex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/tex].

for example lets have two points (9,1) and (8,4) so the value of the slope can be calculated as

4-1 / 8-9

3/-1

= -3

Therefore, the slope of the line passing through the given points is -3

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The legs of a right isosceles triangle are the height and the base of the right isosceles triangle.

It is known that an isosceles triangle is a triangle which has two equal sides.

It is a right isosceles triangle when one side of the triangle is a perpendicular dropped from the vertex and forms a right angle with the base.

As the two sides are equal, the base also turns out to be perpendicular to the height forming a hypotenuse of the third side.

Hence, the two perpendiculars base and height are considered to be the legs of the right isosceles triangle.

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