Find the standard form of the equation of the hyperbola with the given characteristics.
Vertices:
(1, −2), (5, −2);
passes through the point
(−3, 6)

The completely factored form of the expression 4x² + 28x + 49 is (x + 7)(4x + 7) (option a).

In this case, we need to factor the expression 4x² + 28x + 49. One way to do this is by using the quadratic formula, which is a formula used to find the roots or solutions of a quadratic equation.

However, since we are only looking for the completely factored form, we can use a simpler method known as "factoring by grouping." This method involves grouping terms with common factors and factoring them separately.

Starting with the given expression:

4x² + 28x + 49

We notice that the first two terms have a common factor of 4x:

4x(x + 7) + 49

Now, we can factor out the common factor of (x + 7) from the first two terms:

4x(x + 7) + 49 = (x + 7)(4x + 49/7)

Simplifying the expression in the second set of parentheses:

4x + 49/7 = 4x + 7

Hence the correct option is (a).

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Complete Question:

Which is the completely factored form of 4x2 + 28x + 49?

a) (x + 7)(4x + 7)

b) 4(x + 7)(x + 7)

c) (2x + 7)(2x + 7)

d) 2(x+7)(x + 7)

The simplified expression is given as follows:

7/x³.

Where the coefficients are:

The expression in this problem is defined as follows:

x³(1/4 x³)(28x^-9).

To obtain the coefficient c, we must multiply the coefficients of each expression, as follows:

c = 1 x 1/4 x 28

c = 7.

When two terms with the same base and different exponents are multiplied, we keep the bases and add the exponents, hence:

3 + 3 - 9 = -3.

The negative exponents goes in the denominator, hence the simplified expression is given as follows:

7/x³.

Meaning that the exponent is of e = 3.

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Answer:

  √(205)/13

Step-by-step explanation:

You want sec(θ) when cot(θ) = 13/6.

Some calculators can figure this directly as ...

  [tex]\sec{(\cot^{-1}\dfrac{13}{6})}=\boxed{\dfrac{\sqrt{205}}{13}}[/tex]

Others need to make use of the trig relations ...

  tan(θ) = 1/cot(θ)

  sec(θ) = √(tan²(θ) +1)

or

  sec(θ) = 1/cos(θ)

The tangent of the angle is

  tan(θ) = 1/cot(θ) = 1/(13/6) = 6/13.

Then the secant is ...

  sec(θ) = √((6/13)² +1) = √(6² +13²)/13

  sec(θ) = √205/13

A calculator can get there from ...

  sec(θ) = 1/cos(arctan(6/13)) = √205/13

Assuming no withdrawals, the investment will grow to $5960 in approximately 2.63 years.

To answer this problem, use the compound interest formula:

A = (1 + r/n)(n*t).

We're looking for t so we can rearrange the formula to find it:

n = (log(A/P) / log(1 + r/n))

When we substitute the provided values, we get:

12.63 t = (log(5960/5000) / log(1 + 0.048/12))

Hence, assuming no withdrawals, the investment will grow to $5960 in approximately 2.63 years.

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If the daily prices are $ 1.50 for children and $ 2.25 for adults, a total of 284 children and 255 adults swam at the public pool that day.

Let's assume that the number of children who swam in the pool is x, and the number of adults who swam in the pool is y.

From the problem statement, we know that the total number of people who swam in the pool is 539. Therefore, we can write:

x + y = 539

We also know that the total receipts for admission are $1017.00. The price for a child is $1.50 and the price for an adult is $2.25. Therefore, the total revenue can be expressed as:

1.5x + 2.25y = 1017

Now we have two equations with two variables. We can solve for x and y by using any method of solving linear equations. For example, we can use the substitution method to solve for x and y:

x + y = 539 → x = 539 - y

Substituting this value of x into the second equation, we get:

1.5(539 - y) + 2.25y = 1017

Simplifying and solving for y, we get:

y = 255

Substituting this value of y into the equation x + y = 539, we get:

x = 539 - 255 = 284

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If the normally distributed heights have mean as 69, then the probability that he can fit through the doorway without bending is 0.9938.

In "Normal-Distribution", probability refers to the likelihood of a continuous random variable falling within a certain range of values

We first convert the height of the doorway and the height of the male passenger to z-scores by using z = (x - μ) / σ,

where:

x = height (doorway height or male passenger height)

μ = mean of the male height,

σ = standard deviation of male height,

For the doorway height:

⇒ [tex]Z_{doorway}[/tex] = (76 - 69.0)/2.8 = 2.50,

For a male passenger:

⇒ [tex]Z_{male}[/tex] = (x - 69.0)/2.8,

To find the probability that a randomly selected male passenger can fit through the doorway without bending, we need to find the proportion of the "male-population" that has a height less than or equal to the height of the doorway;

⇒ P([tex]Z_{male}[/tex]  ≤  [tex]Z_{doorway}[/tex] ) = P([tex]Z_{male}[/tex]  ≤ 2.50) and

We know that the value of P(z ≤ 2.50) is 0.9938,

Therefore, the required probability is approximately 0.9938.

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The given question is incomplete, the complete question is

An airliner carries 350 passengers and has doors with a height of 76 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in,

If a male passenger is randomly​ selected, find the probability that he can fit through the doorway without bending.

The surface area of the cylinder is 471 mm².

Given is a cylinder of height 5 mm and radius 10 mm, we need to find the surface area of the cylinder,

So, the surface area of the cylinder is = 2π × r (h+r)

= 2 × 3.14 × 5 (5+10)

= 2 × 3.14 × 75

= 471 mm²

Hence, the surface area of the cylinder is 471 mm².

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Answer:

Alex's gross pay for the month is $18/hour x 140 hours = $2520.

The total deductions from his paycheck are:

$180.00 (FICA) + $215.00 (Federal income tax) + $62.00 (State income tax) + $72.89 (Health insurance) + $50.00 (Retirement savings) = $579.89

So Alex's net pay (take-home pay) is:

$2520 (gross pay) - $579.89 (deductions) = $1940.11

To find the percentage of his gross pay that he took home, we can divide his net pay by his gross pay and multiply by 100:

($1940.11 / $2520) x 100 = 77.06%

Therefore, Alex took home 77.06% of his gross pay after deductions.

Let's use the variable "x" to represent the number of 3-inch bolts that James bought.

We know that he bought a total of 16 bolts, so the number of 4-inch bolts he bought would be 16 - x.

The cost of the 3-inch bolts is 36 cents each, so the total cost of the 3-inch bolts would be 0.36x dollars.

Similarly, the cost of the 4-inch bolts is 42 cents each, so the total cost of the 4-inch bolts would be 0.42(16 - x) dollars.

We know that the total cost of all the bolts was $6.00, so we can set up the following equation:

0.36x + 0.42(16 - x) = 6.00

Simplifying and solving for x, we get:

0.36x + 6.72 - 0.42x = 6.00

-0.06x = -0.72

x = 12

Therefore, James bought 12 of the 3-inch bolts.

Answer:

B- 24cm^2

Step-by-step explanation:

for the square part of the shape the length is 7-2=5, 5*4=20

For the triangle, the base is 2, the height is 4, 2*4 is 8, since it’s a triangle the area is divided by 2 so 8/2 is 4.
Adding the areas of the shapes together you get 20+4=24cm^2

Answer:

1) 35%

2) 73%

3) 23%

4) 57%

5) 13%

6) 37%

Step-by-step explanation:

The general solution is to grab the total, which is 100%, and subtract the percentage of the given problems to find the solution. In some problems you have to add some percentages togheder to finish the problem.

So, in the first problem you subtract the percentage of girls to the whole class percentage: 100%-65%=35%.

In the second, whole package minus the red ones: 100%-27%= 73%

In the third, group of children minus swimmers: 100%-78%=22%

In the fourth, 100%-43%= 57%.

In the fifth, there is won percentage, loss percentage, and the remaining percentage you need to add to make 100% is the tie percentage.

So if we do the reverse (subtracting both win and loss percentage to have the tie percentage): 100%-45%-42%=13%.

In the sixth, it's the same thing as the fifth problem: subtracting the basketball and hockey from the whole children percentage that chose the sport: 100%-35%-28%=37%.

I hope this helped

The amount invested at 4% interest rate that is $3900 and

the amount invested at 3% interest rate is $4300.

Last year Sarah invested money in two accounts. The first account had an interest rate of 3% and the second account had an interest rate of 4%. If she invested $400 more in the first account than the second and her total interest income was $285,

the equations are:

       s= f + 400  (1)

      0.03s + 0.04f = 285   (2)

modifying (1) and (2) for simpler calculations

      3s - 3f = 3 * 400 (3)

      3s + 4f = 100 * 285  (4)

by solving eqns (3) and (4) we get

  we get f=3900 and s=4300

Here, f is the amount invested at 4% interest rate that is $3900 and

   s refers to the amount invested at 3% interest rate is $4300.

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i) The after-tax operating cash flows that will be remitted to the parent company each year are USD 73.53, 110.30 and USD 146.34.

ii) The Net Present Value of the project are USD 66.85, USD 91.90 and USD 107.50.

iii) The MNC may want to consider other investment opportunities or reassess the assumptions used in this analysis.

iv) As per the subsidiary, the firm may be better off remitting earnings on a regular basis to minimize exchange rate risk and invest in a diversified portfolio of assets to mitigate other types of risks.

i) To calculate the after-tax operating cash flows that will be remitted to the parent company each year, we need to first calculate the operating cash flows in CAD and then convert them to USD while taking into account the tax on remitted funds. Let's assume the operating cash flows for each year are CAD 100, CAD 150, and CAD 200, respectively.

In year 1, the CAD 100 operating cash flow is remitted to the parent company in USD. Using the exchange rate of C$1.02 per US$, this is equal to USD 98.04. However, the MNC must pay a 25% tax on this amount, so the after-tax cash flow is USD 73.53.

In year 2, the CAD 150 operating cash flow is remitted to the parent company in USD. Using the same exchange rate of C$1.02 per US$, this is equal to USD 147.06. After the 25% tax on remitted funds, the after-tax cash flow is USD 110.30.

In year 3, the CAD 200 operating cash flow is remitted to the parent company in USD. However, the exchange rate has changed to C$1.025 per US$, so the CAD 200 is now equal to USD 195.12. After the 25% tax on remitted funds, the after-tax cash flow is USD 146.34.

Therefore, the after-tax operating cash flows that will be remitted to the parent company each year are USD 73.53, USD 110.30, and USD 146.34, respectively.

ii) To calculate the Net Present Value (NPV) of the project, we need to discount the after-tax operating cash flows back to their present value using a discount rate that reflects the riskiness of the project. Let's assume a discount rate of 10%. Using this rate, we can calculate the present value of each year's after-tax cash flow as follows:

Year 1: USD 73.53 / (1 + 0.10)¹ = USD 66.85

Year 2: USD 110.30 / (1 + 0.10)² = USD 91.90

Year 3: USD 146.34 / (1 + 0.10)³ = USD 107.50

iii) Based on the calculated NPV, the MNC should not accept the project. The negative NPV indicates that the project is not expected to create value for the MNC's investors. Additionally, the after-tax operating cash flows that will be remitted to the parent company each year are not sufficient to compensate for the investment's costs and the 25% tax on remitted funds.

iv) The drawback of Flagstaff Corp.'s approach of reinvesting earnings in Mexican government securities for 10 years before remitting accumulated earnings to the United States is that it exposes the firm to exchange rate risk. Over the course of 10 years, exchange rates between the Mexican peso and the U.S. dollar could fluctuate significantly, potentially eroding the value of the accumulated earnings when they are finally remitted. Additionally, there is no guarantee that the Mexican government securities will continue to provide a high interest rate over the entire 10-year period.

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Answer:

Y-intercept: (0,4)

Slope: 2

Domain: input values

Range: output values

Increasing graph

x-intercept: (-2,0)

Step-by-step explanation:

According to the graph, the y-intercept is (0,4)

the rate of change every x-unit is 2

the x-intercept is (-2,0) as shown in the graph. The rest are just input and output values. Since the slope is positive, the graph is increasing

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R is the set containing the two points (1, 0) and (-1, 0).

In mathematics, a set is a collection of distinct objects, called elements or members, that share a common property. The objects in a set can be anything, such as numbers, letters, shapes, or even other sets. Sets are denoted by braces {} and the elements are separated by commas.

Set theory is an important branch of mathematics and is used in various fields such as algebra, geometry, logic, and computer science. Sets provide a foundation for mathematical concepts such as functions, relations, and number systems. They are also used in the study of probability and statistics, and in the design and analysis of algorithms.

To find R, we need to determine the set of all points (x, y) that satisfy the conditions given:

x² + y² = 1 and |x - y| = 1

Let's consider the two cases of |x - y| = 1:

Case 1: x - y = 1

In this case, we can substitute y = x - 1 into the equation x² + y² = 1 to get:

x² + (x - 1)² = 1

2x² - 2x = 0

2x(x - 1) = 0

So x = 0 or x = 1. If x = 0, then y = -1, which does not satisfy the condition x² + y² = 1. If x = 1, then y = 0, which does satisfy the condition. Therefore, we have one point that satisfies the conditions in this case:

R1 = {(1, 0)}

Case 2: x - y = -1

In this case, we can substitute y = x + 1 into the equation x² + y² = 1 to get:

x² + (x + 1)² = 1

2x² + 2x = 0

2x(x + 1) = 0

So x = 0 or x = -1. If x = 0, then y = 1, which does not satisfy the condition x² + y² = 1. If x = -1, then y = 0, which does satisfy the condition. Therefore, we have one more point that satisfies the conditions in this case:

R2 = {(-1, 0)}

Putting both cases together, we have:

R = R1 ∪ R2 = {(1, 0), (-1, 0)}

Therefore, R is the set containing the two points (1, 0) and (-1, 0).

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the total cost for the company including commission is $15,917

The commission is a sum of money that an employer pays as a fee or a percentage to an employee for completing a certain task or for generating new business by promoting the company's products or services. Incentives or sales commissions are other names for it. In general, anyone can earn a commission; it's not just for salespeople. The commission comes on top of the wage.

The company's policy, which ensures basic compensation and commission over a specific number of deal closures, determines the proportion of commission.

Equity ownership in a corporation is represented by shares. Shares are a type of financial asset that some businesses use to guarantee an equitable dividend distribution of any declared residual profits.

according to question,

total cost of 100 shares = 100*44.41 = $4441

total cost of 600 shares = 600*19.08 = $11448

broker charges for all 700 shares = 700*0.04 = $28

total cost acquired by the company is = $4441+$11448+$28

                                                               = $15917

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The statement that describes the behavior of the function f(x) is: The graph approaches 0 as x approaches infinity.

The correct answer is an option (b)

Consider the function [tex]f(x) = \frac{2x}{1-x^2}[/tex]

Consider the limit of function f(x) as x approaches infinity (x → ∞)

[tex]\lim_{n \to \infty} f(x)\\\\= \lim_{n \to \infty} \frac{2x}{1-x^2} \\\\= \lim_{n \to \infty} \frac{2x/x}{(1-x^2)/x}\\\\ = \lim_{n \to \infty} \frac{-2}{x}[/tex]

= 0

This means that when x approaches infinity the term x² grows rapidly

So, 1 - x², which is in the denominator will decrease, become a large negative more rapid than the x in the numerator.

We can say that the function approaches zero.

Thus the correct statement is: The graph approaches 0 as x approaches infinity.

The correct answer is an option (b)

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The complete question is:

Which statement describes the behavior of the function f (x) = StartFraction 2 x Over 1 minus x squared EndFraction?

a) The graph approaches –2 as x approaches infinity.

b) The graph approaches 0 as x approaches infinity.

c) The graph approaches 1 as x approaches infinity.

d) The graph approaches 2 as x approaches infinity.

Step-by-step explanation:

a it be 3/21 to get red which is 0.143

b to get not red if u don't replace red would be 12/18 so it be 0.6666 but if u replace it then it would be 0.517

Adding and subtracting integrals is a fundamental technique in calculus that allows you to combine multiple integrals into a single expression. The process involves using the properties of integrals and the rules of algebra to simplify the expression.

How to add integrals?

To add two integrals, you simply add the integrands and integrate over the same limits of integration. For example, if you have two integrals, ∫f(x) dx and ∫g(x) dx

The sum of the two integrals would be

∫[f(x) + g(x)] dx

How to subtract integrals?

To subtract two integrals, you subtract the integrands and integrate over the same limits of integration. For example, if you have two integrals ∫f(x) dx and ∫g(x) dx

The difference of the two integrals would be ∫[f(x) - g(x)] dx

It's important to note that when adding or subtracting integrals, you must ensure that the limits of integration are the same. If they are not the same, you must first adjust one or both integrals to have the same limits of integration.

Another important point to keep in mind is that the properties of integrals allow you to simplify expressions before integrating. For example, if you have an integral of the form:

∫f(x) dx + ∫g(x) dx

You can simplify it as ∫[f(x) + g(x)] dx

before integrating.

In conclusion, adding and subtracting integrals is a straightforward process that involves using the rules of algebra and the properties of integrals. By following these techniques, you can simplify complex expressions and solve integration problems more efficiently.

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Correct question is "How to add and subtract integrals?"

Answer:

Step-by-step explanation:

The three true statements about circle Q are:

1. The ratio of the measure of central angle PQR to the measure of the entire circle is One-eighth.

2. The area of the shaded sector is 4 units 2.

3. The ratio of the area of the shaded sector to the area of the circle is equal to the ratio of the length of the arc to the circumference of the circle.

The True statements are:

The ratio of the measure of central angle PQR to the measure of the entire circle is 1/8.

The area of the shaded sector depends on the length of the radius.

The area of the shaded sector depends on the area of the circle.

The measurement of the circle's perimeter, also known as its circumference, is called the circle's boundary. The region a circle occupies is determined by its area.  

First, The measure of the central angle is 45°.  

So, ratio of the central angle to the entire circle

= 45/360

= 9/72

= 1/8

Now, area of shaded sector

A = πr² = π(6²) = 36π

or, 1/8(36π)

= 36π/8

= 18π/4

= 9π/2

= 4.5π square units.  

Since, we use the circle's radius to determine area, the shaded sector's area is dependent on the circle's radius.

Given that the sector is a portion of the circle, its size is also influenced by the size of the circle.

and, The ratio of the shaded sector's area to the circle's area wouldn't be the same as the ratio of the arc's length to the circle's area.

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The balance A for $1900 invested at a rate of 8% for 8 years and compounded n times per year is approximately

$3,538.62, when compounded annually.

$3,601.77 when compounded semi-annually

$3,663.13 when compounded quarterly

$3,710.58 when compounded monthly

$3,748.26 when compounded daily

$3,754.04 when compounded continuously.

We have,

The balance A for $1900 was invested at a rate of 8% for 8 years and compounded n times per year.

A = P(1 + r/n)^(n*t)

where P is the principal amount ($1900 in this case), r is the annual interest rate (8%), t is the number of years (8), and n is the number of times compounded per year.

When compounded annually (n = 1):

A = $1900(1 + 0.08/1)^(1*8) = $1900(1.08)^8 ≈ $3,538.62

When compounded semi-annually (n = 2):

A = $1900(1 + 0.08/2)^(2*8) = $1900(1.04)^16 ≈ $3,601.77

When compounded quarterly (n = 4):

A = $1900(1 + 0.08/4)^(4*8) = $1900(1.02)^32 ≈ $3,663.13

When compounded monthly (n = 12):

A = $1900(1 + 0.08/12)^(12*8) = $1900(1.00666667)^96 ≈ $3,710.58

When compounded daily (n = 365):

A = $1900(1 + 0.08/365)^(365*8) = $1900(1.00021918)^2920 ≈ $3,748.26

When compounded continuously:

A = Pe^(rt) = $1900e^(0.08*8) ≈ $3,754.04

Now,

n     1                2             4           12                365        continuous

A    $3,538.6    $3,601.77  $3,663.13   $3,710.55     $3,748.26   $3,754.04

Therefore,

The balance A for $1900 invested at a rate of 8% for 8 years and compounded n times per year is approximately $3,538.62.

When compounded annually, $3,601.77 when compounded semi-annually, $3,663.13 when compounded quarterly, $3,710.58 when compounded monthly, $3,748.26 when compounded daily, and $3,754.04 when compounded continuously.

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Step-by-step explanation:

The height of the coin, after t seconds, is given by the following equation:

How long will it take the coin to reach the stream?

The stream is the ground level.

So the coin reaches the stream when h(t) = 0.

Multiplying by (-1)

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

This polynomial has roots  such that , given by the following formulas

In this question:

So

Time is a positive measure, so we take the positive value.

It will take 5.61 seconds for the coin to reach the stream.The height of the coin, after t seconds, is given by the following equation:

well, we know the sphere's "great circle" has a circumference of 13mm, so

[tex]\textit{circumference of a circle}\\\\ C=2\pi r ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ C=13 \end{cases}\implies 13=2\pi r\implies \cfrac{13}{2\pi }=r \\\\[-0.35em] ~\dotfill\\\\ \textit{surface area of a sphere}\\\\ SA=4\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=\frac{13}{2\pi } \end{cases}\implies SA=4\pi \left( \cfrac{13}{2\pi } \right)^2 \\\\\\ SA=\cfrac{169}{\pi } \implies SA\approx 53.79~mm^2[/tex]

1. What is 18%? 1500 * .18 = 270
2. Write the equation. 1500 - 270(x) = 50
3. Single out your X by subtracting 1500 from both sides. -250x = -1450
4. X still isn't singled out. Divide by -250. x = 1450/270 (two negatives produce a positive).
5. Simplify. x = 145/27 (fraction), 5.370370 (decimal)

Assuming 18% each year is constant through the months (say, 1.5% decrease each month), your computer will be worth $50 around 5.4 (rounded) years.

the difference between the longest and shortest time Brandon spent reading the book was 3/4 of a day.

How to solve the problem?

To label and plot the data on a dot plot, we need to first determine the range of values. In this case, we have the following data:

1/4, 1/4, 1, 1/4, 1/2, 3/4, 1/2, 1/4

The range of values is from 1/4 to 1, and we can use a dot plot to represent the data. On the dot plot, we will place a dot above each value on the number line. For example, we will place four dots above the value 1/4 since it appears four times in the data.

The dot plot would look like this:

markdown

Copy code

     |

0.2   |   x

     |

0.4   |   x   x   x   x

     |

0.6   |   x   x

     |

0.8   |   x   x   x

     |

1.0   |   x

     |

From the dot plot, we can see that Brandon spent the most time reading the book when he read for the full day, which is represented by the dot at 1. The shortest time he spent reading was when he read for 1/4 of a day, which is represented by the dots at 1/4.

To find the difference between the longest and shortest time, we simply subtract the shortest time from the longest time. In this case, the longest time was 1 day, and the shortest time was 1/4 of a day. So the difference is:

1 - 1/4 = 3/4 of a day

Therefore, the difference between the longest and shortest time Brandon spent reading the book was 3/4 of a day.

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Answer:

Step-by-step explanation:

Dilation by 3 means that you increase the length of each of the sides by 3. Since ED was 2 units long, the plotted dilation of ED is 6 units long. By multiplying each of the lengths of the sides by 3 units, you will be able to draw the whole image.

Answer:

see the attached image

Step-by-step explanation:

For a dilation of scale factor 3, we can multiply each side length by 3.

This polynomial cannot be factored any further using integer coefficients, so our final factored form is: 40w²11 + 16w²6 = 8w²6(5w²5 + 2)

How to solve?

To factor the polynomial 40w²11 + 16w²6, we need to look for common factors in the two terms. We notice that both terms have a factor of 8w²6, so we can factor that out:

40w²11 + 16w²6 = 8w²6(5w²5 + 2)

Now, we have factored out the greatest common factor of the two terms, leaving us with the remaining factor of 5w²5 + 2. This polynomial cannot be factored any further using integer coefficients, so our final factored form is:

40w²11 + 16w²6 = 8w²6(5w²5 + 2)

This form of the polynomial is useful for simplifying expressions, finding roots, and solving equations.

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