What is the relationship between multiplying polynomials and factoring? How are these two concepts related? How can they be used to check your work? Give an example and work it out to explain your reasoning. Be sure to explain how your example shows the relationship you describe.
Things you MUST include:
- A verbal/written explanation that describes the relationship between the two concepts. (3 points)
- An example that shows how multiplying polynomials
Explain how your example shows the relationship. (2 points)

The optimal binary search tree for the given frequency is attached below.

Binary search tree:

Binary search tree refers the value of left node must be smaller than the parent node, and the value of right node must be greater than the parent node.

Given,

Here we have to plot  the optimal binary search tree for the following words, where the frequency of occurrence is in parentheses: a (0.18), and (0.19), I (0.23), it (0.21), or (0.19).

In order to plot the binary search tree, we have to do the following steps,

Then we get the binary search tree like the following.

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The value of (f  o g)(-6) will be;

⇒  (f  o g)(- 6) = - 157

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The values are,

⇒ f (x) = - 4x - 1

And,  g (x) = x² + 3

Now,

Since, The values are,

⇒ f (x) = - 4x - 1

And,  g (x) = x² + 3

Hence, The value of function  (f  o g)(x) is,

⇒  (f  o g)(x) = f (g (x) )

                  = f (x² + 3)

                  = - 4 (x² + 3) - 1

                  = - 4x² - 12 - 1

                  = - 4x² - 13

Thus, (f  o g)(- 6) = - 4 (-6)² - 13

                         = - 4 × 36 - 13

                         = - 144 - 13

                         = - 157

So, The value of (f  o g)(-6) will be;

⇒  (f  o g)(- 6) = - 157

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The expression that is a factor of the polynomial x³ + 2x² - 9x - 18 is (x + 2).

Factoring of a polynomial is the method of breaking the polynomial into a product of its factors.

For example x² - 4 can be broken as (x - 2) and (x + 2).

Therefore, the expression that factor the polynomial x³ + 2x² - 9x - 18 is as follows:

Therefore,

x³ + 2x² - 9x - 18 = (x - 3)(x² + 5x + 6)

Let's break (x² + 5x + 6) down

x² + 5x + 6

x² + 2x + 3x + 6

x(x + 2) + 3(x + 2)

(x + 3)(x + 2)

Therefore, the factors are (x + 3)(x + 2) and (x - 3)

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Answer:

The answer is C, (x+2).

The next time they will meet again is 08:44 (next day)

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

EZPlanes Airways depart = every 264 minutes

Bryanair Airlines depart = every 198 minutes

Both airlines depart = at 06:20

Using the above as a guide, we have the following departure times

EZPlanes Airways depart = 264, 528, 792, 1056, 1320, 1584, 1848, 2112, 2376 and 2640 minutes,

Bryanair Airlines depart = 198, 396, 594, 792, 990, 1188, 1386, 1584, 1782 and 1980 minutes

The common departure time in the above is

Time = 1584 minute

So, we have

Common departure time = 06:20 + 1584 minutes

Evaluate the sum

Common departure time = 06:20 + 26 hours + 24 minutes

Evaluate the sum

Common departure time = 08:44 (next day)

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For the given data

(a) sample mean is 17 and sample standard deviation = 8.91 .

(b) the 95% confidence interval is (9.551 , 24.449)  .

(c) the 90% confidence interval is (11.032 , 22.968) .  .

In the question ,

the data is given as 25, 11, 12, 27,26, 22, 8,5.

So , the number of terms (n) = 8

Part(a)

the sample mean(μ)  is = (25+11+12+27+26+22+8+5)/8 = 17

the standard deviation is calculated as :

= √[(25 - 17)² + (11 - 17)² + (12 - 17)² + (27 - 17)² + (26  - 17)² + (22 - 17)² + (8 - 17)² + (5 - 17)²)]/7

Simplifying the above data further ,

we get

= 8.91

Part(b)

we know that at 95% confidence interval , the critical value is t₀.₀₂₅,₇ = 2.3646

So , the 95% confidence interval is :

= μ ± t₀.₀₂₅,₇*s/√n

= 17 ± 2.3646*8.91/√8

= 17 ± 7.449

So , the interval is ⇒ (9.551 , 24.449) .

Part(c)

we know that at 90% confidence interval , the critical value is t₀.₀₅,₇=1.8946

So , the 90% confidence interval is :

= μ ± t₀.₂₅,₇*s/√n

= 17 ± 1.8946*8.91/√8

= 17 ± 5.968

So , the interval is ⇒ (11.032 , 22.968) .

Therefore ,(a)  the sample mean and standard deviation are 17 and 8.91

(b) the 95% interval is (9.551 , 24.449) .

(c) the 90% interval is (11.032 , 22.968) .

The given question is incomplete , the complete question is

Let the following sample of 8 observations be drawn from a normal population with unknown mean and standard deviation: 25, 11, 12, 27,26, 22, 8,5.

(a) Calculate the sample mean and the sample standard deviation.

(b) Construct the 95% confidence interval for the population mean.

(c) Construct the 90% confidence interval for the population mean.

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The roots of the quadratic equation 3y² + 3(4y² - 2) is ± √(1/7).

A quadratic equation is an algebraic expression in the form of variables and constants.

A quadratic equation has two roots as its degree is two.

Given, 3y² + 3(4y² - 2).

3y² + 12y² - 2.

14y² - 2.

Let 14y² - 2 = 0.

14y² = 2.

y² = (2/14).

y = ± √(2/14) Or ± √(2/2×7) Or ± √(1/7).

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(a) Minimum payment to be paid is $22

(b) Minimum payment to be paid is $35

Simple interest is a quick and simple formula for figuring out how much interest will be charged on a loan.The daily interest rate, the principle, and the number of days between payments are multiplied to calculate simple interest. Although some mortgages employ this calculation approach, this kind of interest typically relates to auto loans or short-term loans.

The formula for simple interest is:

Simple Interest=P×R×T

where:

P=Principal Amount

R=Daily interest rate

T=Number of days between payments​

(a)The minimum monthly payment is calculated by taking 1.5 percent of the outstanding principle

(0.015) ×$ (677+452+139+141) = $21.135

as there is no additional interest due.

The minimum monthly payment required on December 1 is $22, rounded up to the nearest whole dollar.

(b) The minimum monthly payment will be equal to 1.5 percent of the outstanding principle plus the new interest charges.

We will apply the simple interest formula, S.I = p×r×t, to get his new interest costs.

Because Dec contains 31 days, the

Principle amount, p = (677+452+139+141) - 300 = 1109,

the rate, r = 0.05163% each day

Time, t = 31 days.

S.I =p×r×t=1109×0.0005163×31= $17.75

0.015×1109 = 16.64, which is 1.5 percent of the outstanding amount.

To reach $17.75 + $16.64 = $34.39, we must now add the additional interest to 1.5 percent of the outstanding principal.

The minimum monthly payment required on January 1 is $35, rounded up to the next whole dollar.

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Answer:

1: 50

2: 1

3: 2

4: 36

5: 0

6: -12

7: -2

8: 3

Step-by-step explanation: Use PEMDAS

P: Parentheses

E: Exponents

M: Multiplication

D: Division

A: Addition

S: Subtraction

Hi,

I hope you and your family are doing well!

An angle is coterminal with another angle if it has the same terminal side as the other angle, but may have a different degree measure. In other words, coterminal angles are angles that are measured around the same point on the unit circle, but may start at different points.

To find the angle between 0 and 2π in radians that is coterminal with the angle -25π/6, we can add or subtract multiples of 2π to -25π/6 until we get an angle between 0 and 2π.

For example, adding 2π to -25π/6 gives us:

-25π/6 + 2π = -π/6 + 2π = 7π/6

This angle is between 0 and 2π, so it is coterminal with -25π/6.

Alternatively, subtracting 4π from -25π/6 gives us:

-25π/6 - 4π = -29π/6 = -π/2

This angle is also between 0 and 2π, so it is coterminal with -25π/6.

Therefore, the angle between 0 and 2π in radians that is coterminal with -25π/6 is either 7π/6 or -π/2.

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Happy Holidays & New Year!

The area under the graph using first two and then four rectangles f(x) = x³ between x = 2 and x = 4 is 0.2421875.

Here a=0, b=1 and n=2( two rectangles), so

Δx = (1 - 0)/2

Δx = 1/2

Δx = 0.5

M2 = (0.5) * ( f((0+0.5)/2)) + f((0.5+1)/2)) )

= (0.5) * ( f(0.25) + f(0.75) )

Using  f(x) = x^3

=(0.5) * ( 0.015625 + 0.421875 )

=0.21875

Here a=0, b=1 and n=4( four rectangles), so

Δx = (1 - 0)/4

Δx = 1/4

Δx = 0.25

M4 = (0.25) * ( f((0+0.25)/2)) + f((0.25+0.5)/2)) + f((0.5+0.75)/2)) + f((0.75+1)/2)) )

=(0.25) * ( f(0.125) + f(0.375) + f(0.625) + f(0.875) )

Using  f(x) = x^3

= (0.25) * ( 0.001953125 + 0.052734375 + 0.244140625 + 0.669921875 )

= 0.2421875

Hence,  the area under the graph using first two and then four rectangles f(x) = x³ between x = 2 and x = 4 is 0.2421875.

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The value equivalent to   [tex](64)^{1/4}[/tex] is [tex]2\sqrt[4]{4}[/tex].

An expression or mathematical expression is a finite arrangement of symbols that are well-formed in line with context-dependent criteria. To help define the logical grammar and order of operations, mathematical symbols can be used to represent variables, operations, functions, brackets, punctuation, and grouping.

A formula and an expression are often distinguished by authors, with the former designating a mathematical entity and the latter a statement about such a thing.

But in contemporary mathematics, and particularly in computer algebra, formulas are seen as expressions that can be evaluated as true or false, depending on the values given to the variables present in the expressions.

Given expression is  [tex](64)^{1/4}[/tex]

The number 64 can be written as 2 × 2 × 2 × 2 × 2× 2 =2⁶

So, the above given expression becomes,

[tex](64)^{1/4}[/tex]=[tex](2^{6}) ^{1/4}[/tex]

2⁶=2⁴.2²

So,

[tex](2^{6}) ^{1/4}[/tex]=[tex](2^{4} 2^{2})^{1/4}[/tex]

By simplifying we get,

[tex](64)^{1/4}[/tex]=[tex]2\sqrt[4]{4}[/tex]

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Answer:

The possible values of k are -0.56, -3.56, -2 and -1.

Step-by-step explanation:

We have the equation of the parabola, .

Substituting x=0, we get that  i.e. y = -1.5k

So, the y-intercept is (0,-1.5k).

Thus, the height of the triangle becomes |-1.5k| = 1.5k

Again, substituting y=0,  i.e.  i.e. x=3 and x=-k

So, the x-intercepts are (3,0) and (-k,0).

Thus, the base of the triangle is 3+k if k>-3 or -3-k if k<-3.

We see that 'k' cannot be 0 or -3 as if k=0, then height = 0, which is not possible. If k= -3, then the base = 0, which is also not possible.

As, area of a triangle = .

Substituting the values, we get,

1.5=

i.e.

i.e. k = -0.56 and k = -3.56

or

1.5=

i.e.

i.e. k = -2 and k = -1.

Thus, the possible values of k are -0.56, -3.56, -2 and -1.

The total cost of installing a fence around all sides of the rectangular yard would be $38250.

What is the area of the rectangle?

For the dimensions of the rectangle of length 'l' and breadth 'b', the area can be computed by using the formula,

           Area = Length ×  Breadth

Given data:

The length of the yard is 90 feet. and the breadth is 50 feet.

Area of the yard = length * breadth

                           = 90 * 50

                           = 4500 square foot

Now per square foot, the cost is $8.50 so the total cost of installing a fence around all sides of the yard would be = 4500 * 8.5

                                                                          = $38250.

Hence, the total cost of installing a fence around all sides of the rectangular yard would be $38250.

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Answer:

64/100

Step-by-step explanation:

Answer: 16/25or64/100

Step-by-step explanation: 64/100 and divide it by 4 . To change a percent to a fraction, put the percentage over 100 (after removing the % sign) and simplify if necessary

The required solution (1, 4) of a linear system of the equation is determined by the elimination method.

Given the linear system of equations:

8x - 2y = 0 or 4x - y = 0 …(i)

-6x + 14y = 50 or -3x + 7y = 25 …(ii)

Equations (i) and (ii) constitute a system of two first-degree equations in the two variables x and y.

So, we have to find out the value of 'x’ and 'y’.

By multiplying 7 by equation (i) and adding both equations

28x - 7y -3x + 7y = 25

25x = 25

x = 25/25

x = 1

Substitute the value of x = 1 in the equation (i), and solve for y

4(1) - y = 0

y = 4

Hence, the required solutions are x = 1 and y = 4.

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Using Pythagorean Theorrem the length of AE = 46 units

What is a Right Angle Triangle?

A right triangle or right-angled triangle, or more technically an orthogonal triangle, formerly known as a rectangled triangle, is a triangle with one right angle, i.e. two perpendicular sides. Trigonometry is founded on the relationship between the sides and other angles of a right triangle.

Solution:

In Triangle ABC using Pythagorean Theorem

AC = √(21*21 + 20*20)

AC = √841

AC = 29

In Triangle CDE using Pythagorean Theorem

CE = √(8*8 + 15*15)

CE = √289

CE = 17

From the figure we can say that AE = AC + CE

AE = 29 + 17

AE = 46 units

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The required distance between the points  (−1,6) and (−1,7). is 1 unit.

Given that,
using the distance formula to evaluate the distance between the points  (−1,6) and (−1,7).

Distance is defined as the object traveling at a particular speed in time from one point to another.

Here,
The distance formula is given as,
D = √[[x₂ - x₁]² + [y₂+ - y₁]²]
Substitute the values in the above equation,
D = √[[-1 + 1]² + [7 - 6]²]
D = √[0 + 1]
D = 1

Thus, the required distance between the points  (−1,6) and (−1,7). is 1 unit.

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Answer:

Step-by-step explanation:

First, you will multiply 2 times 4,374 and get 8,748 then you will divide 8748 by 7 which is a week, and get 1249.7 then you will divide it by 356 which is a year approximate, and get 10.51 days.

a. Using linear equations in one variable we get the number of songs at which both companies would charge the same amount is 100 songs.

b. Company B would be a better choice as the total charges for 150 songs would be $127.5.

Explain linear equations in one variable.

A straight line with only one variable is a linear equation. the single variable's power is 1. Simple algebraic methods are used to solve these equations, which have the form ax+b=0.

a) Assume the number of songs downloaded is 'x'.

Now according to the question,

Total charges by company A is 15+0.75x

Total charges by company B is 0.90x

For equal charges, both must be equal,

Hence equating the 2 expressions we get,

15+0.75x=0.90x

15=0.15x

x=100

Hence total number of songs to reach the charge of the same amount is 100.

b)Let us take 150 songs from company A, we get the total charge to be

15+0.75*150=$127.5

Taking for company B, we get

0.90*150=$135.

We see that for 150 songs, the charges of company A is less compared to company B. Therefore I would choose company B over company A.

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Answer

hello I have provided a picture of the answer

Step-by-step explanation:

ur welcome and have a nice day

In 11.61 years the given principal amount of $ 900 will be double.

Compound interest is interest charged on a loan and is determined by both the principal amount and the rate of interest compounding per year.

The formula for the Amount when compound interest is calculated is given by

P = Principal amount

r = Rate of interest

n = Number of compounds

t = Time period in years

Here from the given data,

Principal amount = $ 900

Rate of interest = 6% = 6/100 = 0.06

Number of compounds = 12    [ ∵ compounded monthly]

Here we need to find the time period

Assume that after 't' years the amount will be doubled

From the given data and formula,

Amount, A = 900(1+0.06/12)^12t        

= 900(1+0.06/12)^12t

= 900(1 + 0.005)^12t

= 900 (1.005)^12t  

As we assumed after 't' years the amount is double

=> 900 (1.005)^12t  = 2 × 900

=> (1.005)^12t  = 2    

Applying log on both sides

=> log (1.005)^12t  = log 2

=> 12t log(1.005) = log 2

=> 12t = log 2/log(1.005)

[ we can use a calculator for log (1.005) and log 2]

=> 12t = 0.30108/ 0.00216  

=> 12t = 139.888

=>  t =  11.61

Therefore,

In 11.61 years the given principal amount of $ 900 will be double.

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The angles of the cyclic quadrilateral and the arc angle is as follows:

A cyclic quadrilateral is a quadrilateral drawn inside a circle. In other words, a cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle.

A cyclic Quadrilateral's opposite angles add to 180°.

Therefore,

∠PSR = 180 - 93 = 87 degrees.

An inscribed angle is one-half of the measure of the arc that it intercepts.

Therefore,

87 = 1 / 2 arc PQR

cross multiply

arc PQR = 87 × 2

arc PQR  = 174 degrees

Hence,

arc QR = 174 - 126

arc QR = 48 degrees

arc RS = 360 - 90 - 174

arc RS = 96 degrees.

∠QPS = 1 / 2 (96 + 48)

∠QPS = 1 / 2 (144)

∠QPS = 144 / 2

∠QPS = 72 degrees

∠SRQ = 180 - 72

∠SRQ = 108 degrees

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Answer:    

   48°

   96°

    72°

    87°

   108°

Step-by-step explanation:

   arc QR = 48°

   arc RS = 96°

   ∠QPS = 72°

   ∠PSR = 87°

   ∠SRQ = 108°

Answer:

4/5

Step-by-step explanation:

5/6q-1/3q=2/5

We change the coefficient’s denominators to 6 so we can subtract them

5/6q-2/6q=2/5

3/6q=2/5

1/2q=2/5

Now divide both sides by 1/2 or multiply both sides by 2

1/2q=2/5

/1/2.  /1/2

q=2/5/1/2

q=2/5*2/1

q=4/5

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By using the concept of algebraic expression, it can be calculated that

The number that makes the expressions equivalent is -0.7

What is algebraic expression?

Algebraic expression consist of variables and numbers connected with addition, subtraction, multiplication and division.

The given algebraic expression is

−1.1−1.4m + 0.4

On simplifying −1.1−1.4m + 0.4

−1.1−1.4m + 0.4

-0.7 - 1.4m

So the number that makes the expressions equivalent is -0.7

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Answer:

1.25 es la respuesta

Step-by-step explanation:

Answer:

Step-by-step explanation:

Given:

Each picture costs $7.50

One time shipping cost = $3.25

Total cost of the order = $85.75

To FInd:

An equation to determine the number of pictures ordered.

Solution:

Define x:

Let x be the number of pictures bought.

Let y be the total amount of money collected.

Form the equation:

The $3.25 one time shipping cost is a fixed cost.

The $7.50 is a variable cost depending on the number of pictures ordered.

Therefore the equation:

⇒ y = 3.25 + 7.5x

Find the number of pictures bought.

The total cost is $85.75

3.25 + 7.5x = 85.75

7.5x = 82.50

x = 11

Answer: The equation is y = 3.25 + 7.5x and Todd bought 11 pictures.

The equation of the line that passes through the point (3,4) and has an undefined slope is y = 3

An undefined slope is an infinite slope. This is the slope of a vertical line. The x values never changes no matter the changes in the y values.

The equation of a line in slope intercept form can be represented as follows:

y = mx + c

where

m = slope

c = y-intercept

Since the slope, m is undefined, the expression that has an undefined slope is a vertical line

Therefore,

y = 3

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Answer:

The answer is 62 and 86

Step-by-step explanation:

Mean

(50 + 70 + 58 + 62 + 66 + 53 + 76) ÷ 7

435/7 = 62.1 ≈ 62

Variance

(50 - 62)² + (70 - 62)² + (58 - 62)² + (62 - 62)² + (66 - 62)² + (53 - 62)² + (76 - 62)²

(144 + 64 + 16 + 0 + 16 + 81 + 196)/(7 - 1)

517/6 = 86

Thus, The mean of the given data is 62 and Variance is 86