That, given an array a of n integers, returns the smallest positive integer (greater than 0) that does not occur in a. For example, given a = [1, 3, 6, 4, 1, 2], the function should return 5

The partial fraction expansion of (x³ + 4)/ ((x² - 1)(x² + 3x + 2)) is -3/ 4(x + 1) + 5/12(x - 1) + 4/3(x + 2) - 3/ 2(x + 1)². So the coefficients A = -3/4, B = 5/ 12, C = 4/3 and D = -3/2.

A partial fraction expansion is a way of expressing a rational function (a function that can be written as the ratio of two polynomials) as the sum of simpler fractions, each with a numerator that is a constant or a simple polynomial. The process of finding the partial fraction expansion of a rational function is also known as partial fraction decomposition.

The denominator is

(x² - 1)(x² + 3x + 2) = (x² - 1)(x² + x + 2x + 2)

(x² - 1)(x² + 3x + 2) = (x + 1)(x - 1)(x + 1)(x + 2)

So by  partial fraction expansion

(x³ + 4)/ ((x² - 1)(x² + 3x + 2)) = (x³ + 4)/ ((x + 1)(x - 1)(x + 1)(x + 2))

(x³ + 4)/ ((x + 1)(x - 1)(x + 1)(x + 2)) = A/ (x + 1) + B/(x - 1) + C/(x + 2) + D/ (x + 1)²

A(x - 1)(x + 1)(x + 2) + B(x + 1)²(x + 2) + C(x + 1)²(x - 1) + D(x - 1)(x + 2) = x³ + 4

(A + B + C)x³ + (2A + 4B + C + D)x² + (-A + 5B - C + D)x + (-2A + 2B - C - 2D) = x³ + 4

Thus from coefficients of x³, x², x and 1,

A + B + C = 1

2A + 4B + C + D = 0

-A + 5B - C + D = 0

-2A + 2B - C - 2D = 4

Solving

A = -3/4, B = 5/ 12, C = 4/3 and D = -3/2

--The question is not clear, the question is given below--

"Enter (A,B,C,D) in order below if A, B, C, and D are the coefficients of the partial fractions expansion of

[tex]$$12\cdot\frac{x^3 + 4}{(x^2-1)(x^2 +3x+ 2)}[/tex]  "

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The number that belongs to both sequences is 26.

The formula known as the nth term allows us to locate any term in a series. The letter "n" stands for "number."

By changing the term number's value, we can create a series that uses the nth term (n).

To find the 10th term we would follow the formula for the sequence but substitute 10 instead of ‘n‘; to find the 50th term we would substitute 50 instead of n.

We have two sequences, let's call one as A and the other as B.

The n-th term of sequence A is written as:

aₙ = 3*n^2 - 1

the nth term of sequence B is written as:

bₙ = 30 - n^2

We want to find a term that belongs to both sequences, (it can be for different integers, we can use n for sequence A and x for sequence B)

Then we want to find:

aₙ = bₓ

where n and x are integer numbers.

Then we will heave:

3*n^2 - 1 = 30 - x^2

To find the pair, we could isolate one of the variables, then input different integers in the other variable and see if the outcome is also an integer.

Let's isolate n.

3*n^2 = 30 - x^2 + 1

3*n^2 = 31 - x^2

n^2 = (31 - x^2)/3

n = √(  (31 - x^2)/3)

Now let's input different values for x, and see if the outcome is also an integer, notice that x is in a negative term inside a square root, then we have only a few values of x such that the equation can be true.

Then let's start with x = 1.

n(1) = √(  (31 - 1^2)/3) = √(30/3) = √10

We know that √10 is not an integer.

now with x = 2,

n(2) = √(  (31 - 2^2)/3)  = √( (31 - 4)/3) = √(27/3) = √9 = 3

then if x = 2, we have n = 3.

Both of them are integers, then we get:

a₂ = 3*(3)^2 - 1 = 27 - 1 = 26

b₃ = 30 - 2^2 = 30 - 4 = 26

Therefore, The number that belongs to both sequences is 26.

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Answer:

f(-7) = -33

Step-by-step explanation:

f(x) = − x² - 4x – 12

f(-7) = ?

We put -7 in for x and solve

f(-7) = -(-7)^2 -4(-7) - 12

f(-7) = -(49) + 28 - 12

f(-7) = -49 + 28 - 12

f(-7) = -33

The area, in square units, of the region where the two triangles overlap is found as 1 square units.

We can determine the equation for each line with in diagram using its coordinates.

We can determine the coordinates for each point. (1/4,3/4) is one of the intersections.

d² = 2² + 2² = 8

A₁ = (1/3d + 1/6d )/2 * 1/4d

A₁ = 1/4d * 1/4d

A₁ = 1/ 16 d²

Now,

A = 2A₁

A = 2.(1/16)d²

A = 1/8d²

A = 1/8 *8

A  = 1

Thus, the area, in square units, of the region where the two triangles overlap is found as 1 square units.

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Answer:

1 square unit

Step-by-step explanation:

No, 16x2 40x 25 is not a perfect square trinomial.

Perfect squares are numbers that are the result of multiplying a number by itself. Examples of perfect squares include 4, 9, 16, 25, 36, and 49. Perfect squares are also known as "square numbers" or "whole numbers." Perfect squares are not just limited to whole numbers; some decimals and fractions can also be perfect squares.

A perfect square trinomial is an expression of the form a2 + 2ab + b2, where a and b are constants. 16x2 40x 25 does not fit this form, and is therefore not a perfect square trinomial.

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Answer:

12%

Step-by-step explanation:

The volume of the box is 58 cm³. Therefore, the units of the answer are cm³.

Volume is the measure of the amount of 3-dimensional space occupied by an object or substance. It is usually measured in cubic units, such as cubic centimeters (cm3) or cubic meters (m3). Volume is an important concept in mathematics, physics, chemistry, and engineering, and is used to calculate the amount of material needed for a given project.

The volume of the box can be calculated by multiplying the area of the base with the height of the box. The area of the base is 58 cm² and the height of the box is x - 2 cm. Therefore, the volume of the box can be expressed as:
Volume = 58 cm² x (x - 2 cm)
     = 58 cm³
The volume of the box is 58 cm³. Therefore, the units of the answer are cm³.

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Answer:

894 mi²

Step-by-step explanation:

the area of parallelogram= base x height= 24mi x 37.25mi = 894 mi²

Answer:

The hundreds place

Step-by-step explanation:

143, first digit is , 1 is in the hundreds place

By using formula, P(A n B) = 0.346. Rounded to the nearest thousandth, P(A n B) = 0.346

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. In between 0 and 1, the closer the probability is to 1, the more likely the event is to occur. The probability of an event is usually represented by the letter P, followed by the event in parentheses.

Rounding off is a mathematical operation that is used to reduce the number of digits in a number while maintaining its value within a certain degree of accuracy. When we round off a number, we look at the digit that is immediately to the right of the last digit that we want to keep. If the number is less than 5, we keep the last digit as it is, but if the number is greater than or equal to 5, we add 1 to the last digit.

We can use the formula of conditional probability to find the value of P(A n B).

P(A n B) = P(A U B) - P(A) - P(B) + P(An B)

We know that:

P(A U B) = P(A) + P(B) - P(A n B)

So we can substitute this in the first equation:

P(A n B) = P(A U B) - P(A) - P(B) + P(An B)

P(An B) = P(A U B) - P(A) - P(B) + P(An B)

Now we can substitute the given values:

P(An B) = 0.366 - 0.32 - 0.2 + P(An B)

P(An B) = 0.346 + P(An B)

To find P(An B) we can subtract 0.346 from both sides of the equation:

P(An B) = P(An B) - 0.346

P(An B) = P(An B) - 0.346

So, P(An B) = 0.346

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Answer:

[tex] \displaystyle{ y= - \dfrac{ 3}{2} x+ 1}[/tex]

Step-by-step explanation:

Slope-intercept is in form of y = mx + b where m is slope and b is y-intercept. We simply solve for y in term of x and then we will obtain the slope-intercept form.

Given the linear equation:

[tex] \displaystyle{3x + 2y = 2}[/tex]

First, subtract both sides by 3x:

[tex] \displaystyle{3x + 2y - 3x= - 3x + 2} \\ \\ \displaystyle{2y = - 3x + 2}[/tex]

Then divide both sides by 2:

[tex] \displaystyle{ \dfrac{2y}{2} = \dfrac{ - 3x + 2}{2}} \\ \\ \displaystyle{ y = \dfrac{ - 3x + 2}{2}}[/tex]

Separate fractions (Simplify):

[tex] \displaystyle{ y= \dfrac{ - 3x}{2} + \dfrac{2}{2}} \\ \\ \displaystyle{ y= - \dfrac{ 3}{2} x+ 1}[/tex]

And we have finally converted the equation to slope-intercept form.

Answer:  15cm2

Step-by-step explanation:

5cm+4cm+6cm=15cm2

Answer:

X = 1 and below

Step-by-step explanation:

since X is less than 2.

the value of X will be 1 and below.

Answer:

Alex deleted 37.5 selfies

Step-by-step explanation:

Let X be the number of selfies that Vic deleted.

Alex deleted 3X selfies.

After deleting selfies, Alex has 90-3X selfies left.

Vic has 75-X selfies left.

Since Alex has half as many selfies as Vic, we can set up the equation: 90-3X = (75-X)/2

Multiplying through the parentheses on the right side, we get: 90-3X = 75-X/2

Combining like terms, we get: 2X = 15

Dividing both sides by 2, we get: X = 7.5

Since X represents the number of selfies that Vic deleted, Vic deleted 7.5 selfies.

Since Alex deleted 3 times as many selfies as Vic, Alex deleted 37.5 = <<37.5=22.5>>22.5 selfies.

The number of selfies deleted by Alex is 51.

An equation is a mathematical statement that shows that two mathematical expressions are equal.

Given that Alex took 90 selfies and Vic took 75 selfies. Today, they each decided that some of these selfies were not very good and deleted them. Alex deleted three times as many selfies as Vic. Now, Alex has half as many selfies as Vic.

Let Vic deleted x selfies and number of selfies deleted by Alex is 3x,

Therefore,

Since, Alex has half as many selfies as Vic, we can set up the equation: 90-3x = (75-x)/2

180 - 6x = 75 - x

5x = 105

x = 17

3x = 51

Hence, the number of selfies deleted by Alex is 51.

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The height of vase b is 34.56 cm.

In mathematics, volume is the space taken by an object.

V=pi*r²*h

Here, given that,

vase A has height 20cm =H

and volume 1500^3. =V

And, A & B similar vases.

Again, vase B has volume 2592cm^3 =v

let, h, the height of vase b.

both radius are equal

so, we get,  V/pi*H = v/pi*h

i.e. 1500/20=2592/h

so, h= 34.56 cm

Hence, the height of vase b is 34.56 cm.

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Answer:

9 in.

Step-by-step explanation:

the sum of two sides in a triangle is always longer than the 3rd side. so 9 is the only answer that fits this condition.

Answer:

"If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular."

Step-by-step explanation:

Answer:

It will take him 9 weeks to be able to buy the bike.

Cristiano is saving money to buy a bike that costs $189. He saved $99 so far. He plans on saving $10 each week. The correct answer is 9 weeks.

So, we have:

- Cost of the bike = $189

- Money saved = $99

- Money saving each week = $10

We are trying to find how many weeks will he have enough money to the bike.

So, we are going to multiply and add the numbers to find are answer:

→ (n × $ 10) + $ 99 = $ 189

→ n × $ 10 = $ 90 n = $ 90 $ 10 n=9 weeks.

Therefore, Cristiano is saving money to buy a bike that costs $189. He saved $99 so far. He plans on saving $10 each week. The correct answer is 9 weeks.

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Answer:

d = 117t

Step-by-step explanation:

The equation is

y = mx + b

Here m is the function's slope, and b is the y-intercept.

We know

As the car moves at a constant speed passes a timing device at t=0.

After 9 seconds, the car traveled 1053ft.

We have to find the slope

Slope = rise/run or (y2 - y1) / (x2 - x1)

We have the slope is

m = 1053/ 9 = 117

So, the equationn is d = 117t

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A guy wire is attached to an upright pole 6 meters above the ground. If the wire is anchored to the ground 5 meters from the base of the pole 7.81 m long is the wire. This can be solved by using the concept of Pythagoras theorem.

The widely accepted geometric principle known as the Pythagorean Theorem states that the square on the hypotenuse of a right triangle equals the sum of the squares on its legs (the side opposite the right angle).

The hypotenuse side of a right-angled triangle's square is equal to the sum of its other two sides, according to Pythagoras' Theorem. The perpendicular, base, and hypotenuse of this triangle are its three designated sides.

Let, assume ΔABC is a right-angled triangle

where, AC = 6m be the pole.

AB = 5m is the length of a guy wire and is attached to stake B

By Pythagoras theorem,

BC² = AB² + AC²

or, BC² = (5)² + (6)²

or, BC² = 36 + 25

or, BC =  √61 m

or, BC = 7.81 m

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The general rule is power of power. It can be represented by (x^m)^n = x^mn.

What is an exponent?

The way of representing huge numbers in terms of powers is known as an exponent. Exponent, then, is the number of times a number has been multiplied by itself.

Now complete the given table:

Problem to           First repeated        second repeated               Power of

simplify                 multiplication         multiplication              the form of a^n

(2²)³                         2²×2²×2²               2×2×2×2×2×2                           2⁶

(5³)⁴                     5³ × 5³ × 5³ × 5³      5×5×5×5×5×5×5×5×5×5×5×5    5¹²

(7³)⁴                     7³ × 7³ × 7³ × 7³      7×7×7×7×7×7×7×7×7×7×7×7        7¹²

((3x)²)³                   (3x)²×(3x)²×(3x)²    3x×3x×3x×3x×3x×3x                    (3x)⁶

(4y²)³                   (2y)²×(2y)²×(2y)²    2y×2y×2y×2y×2y×2y                    (2y)⁶

The general rule is  (x^m)^n = x^mn. The name of the rule is power of power.

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The directions of XYZ in graph are left and right, top and bottom and outward from the screen respectively.

The term graph is known as a pictorial representation or a diagram that represents data or values in an organized manner.

Here we have to find the directions of XYZ on the graph.

Here we have know that X-Y-Z matrix is a three-dimensional structure whereby the x-axis and y-axis denote the first two dimensions and the z-axis is the third dimension.

And then in a graphic image, then the x denotes width, y denotes height and the z represents depth.

The directions of the x axis is in the plane of the screen and is positive toward the right and negative toward the left. Where as the directions of the y axis is in the plane of the screen and is positive toward the top and negative toward the bottom.

Finally the directions of the z axis is perpendicular to the screen or keyboard, and is positive extending outward from the screen.

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Yes, a 45 45 triangle is an isosceles triangle, meaning that two of its sides are equal in length.

A 45 45 triangle is a triangle in which two of its angles are 45 degrees. This type of triangle is also known as an isosceles triangle because it has two sides of equal length. To determine whether a triangle is isosceles, we need to measure the lengths of its sides. In a 45 45 triangle, both sides are equal in length as the angles are equal. Therefore, a 45 45 triangle is an isosceles triangle. The triangle also has two acute angles (less than 90 degrees) and one obtuse angle (greater than 90 degrees). This type of triangle is a special type of triangle and is often used in geometry and trigonometry.

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Three straight sides and three angles make up the shape of a triangle. The angles of a triangle can either be acute, right, or obtuse. An angle with three acute angles is a triangle where all three angles measure less than 90 degrees.

A triangle is a three-sided shape with three angles. The sum of the internal angles of a triangle is always 180 degrees. An angle is the amount of turn or change in direction between two lines. There are three types of angles, acute, right and obtuse. An acute angle is an angle which measures less than 90 degrees. A right angle is an angle which measures exactly 90 degrees, and an obtuse angle is an angle which measures more than 90 degrees. Therefore, an angle with three acute angles is a triangle where all three angles measure less than 90 degrees. A triangle with three acute angles is also known as an acute triangle. All three angles must add up to 180 degrees, so the measure of each angle must be less than 60 degrees.

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Step-by-step explanation:

continuation of the previous answer:

now, for the final step angie should use the change of base formula in order to make the logarithmic expression simpler to analyze.

[tex]x = \frac{ log(6) }{ log(23) } [/tex]

after applying this formula Angie has successfully solved her exponential equation.

Answer:

Angie should use the change of base formula to make this equation faster. Which would be x= log(6)/log(23). Then, after using this formula and just plugging it into your calculator, Angie should have successfully answered her question. 

Step-by-step explanation:

I took the exam

Answer:

x = 2

Step-by-step explanation:

4x + 3 = 11

(4x + 3) - 3 = 11 - 3

4x = 8

4x/4 = 8/4

x = 2

As you can see, to isolate x, we must do the opposite operations to both sides.

The interquartile range of the data decreases when the eighth day's amount is included in the data is true.

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

Joanna set a goal to drink more water daily.

The number of ounces of water she drank each of the last seven days is

60, 58, 64, 64, 68, 50, 57

On the eighth day, she drinks 82 ounces of water.

Raw data:      [60,58,64,64,68,50,57,82],

Sorted data:  [50,57,58,60,64,64,68,82]

Sample size = 8 (even)

mean  = 62.875

median  = (60+64)/2 = 62

1st quartile   = (57+58)/2 = 57.5

3rd quartile  = (64+68)/2 =  66

IQR = 66 - 57.5 = 8.5

Data without the eight day's measurement.

Raw data:      [60,58,64,64,68,50,57]

Sorted data:  [50,57,58,60,64,64,68]

Sample size = 7 (odd)

mean            = 60.143

median         = 60

1st quartile   = 57

3rd quartile = 64

IQR = 64 -57 = 7

Hence, the interquartile range of the data decreases when the eighth day's amount is included in the data is true.

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Answer: It's just

The interquartile range of the data decreases when the eighth day's amount is included in the data.

Step-by-step explanation: The question asks you to select ALL correct answers, but there is only one correct answer.

The number of groups of like terms is 3

An algebraic expression is defined as an expression made up of terms, variables, coefficients, constants, and factors.

These algebraic expressions are also composed of mathematical operations, such as;

A term can be described a single variable, number or numbers and variables that are multiplied together.

From the information given, we have the terms;

3x¹³, 9x¹⁰, -1x¹³, -16x¹³, 3x¹⁰, 3¹³, 10x¹⁰, 13x¹⁰

The similar terms are;

3x¹³,  3x¹⁰,  3¹³

Hence, the number is 3

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The 30th term of the arithmetic sequence is 265

What is an arithmetic sequence in math?

This is an arithmetic sequence since there is a common difference between each term. In this case, adding 9 to the previous term in the sequence gives the next term. In other words,   [tex]a_{n}[/tex]= [tex]a_{1}[/tex]+ d ( n - 1 )

Arithmetic Sequence: d = 9

This is the formula of an arithmetic sequence.

[tex]a_{n}[/tex] =  [tex]a_{1}[/tex] + d ( n - 1 )

Substitute in the values of   [tex]a_{1}[/tex] = 4 and  d = 9

 [tex]a_{n}[/tex]= 4 + 9 ( n - 1 )

[tex]a_{n}[/tex]= 4 + 9n - 9

[tex]a_{n}[/tex] = 9n - 5

.30th term =   [tex]a_{n}[/tex] =   [tex]a_{1}[/tex]+ d ( n - 1 )

=  [tex]a_{30}[/tex]= 4 + 9 ( 30 -1 )

   [tex]a_{30}[/tex]=   4 + 9 x 29

  [tex]a_{30}[/tex]= 265

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