8 Which expression could NOT be used to calculate the
perimeter of the rectangle below?
x + 10 2x
A: 2x(x + 10)
B: 2(2x + x + 10)
C:2(2x) + 2(x+10
D: 2x + 2x + x + 10 + x + 10

Answer:

Step-by-step explanation:

I think 3

The total number [y] of bracelets she can make will be -

y = 12/x.

Given is that Anita is making bracelets. She has 12 feet of string.

Assume that the length of each bracelet be [x] ft. Then, the total number [y] of bracelets she can make will be -

y = 12/x

Therefore, the total number [y] of bracelets she can make will be -

y = 12/x.

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The trinomial expression that represents C is

C = -3x² - 18x + 19

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

C = G - 3F ______(1)

F = 2x² + 6x - 5

G = 3x² + 4

Substituting F and G in (1)

C = 3x² + 4 - 3(2x² + 6x - 5)

C = 3x² + 4 - 6x² - 18x + 15

C = -3x² - 18x + 19

Thus,

The trinomial expression is

C = -3x² - 18x + 19

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It is proven below that

\(o(x) = o(y^(-1)xy)\)

for all x, y in every group G.

To prove that

\(o(x) = o(y^(-1)xy)\)

for all x, y in every group G, show that the order of the element x is equal to the order of the conjugate

y⁻¹xy

Let's proceed with the proof:

1. Let x, y be arbitrary elements in the group G.

2. Consider the element

y⁻¹xy

3. To show that

\(o(x) = o(y^(-1)xy),\)

we need to prove that

(y⁻¹xy)ⁿ = e

(the identity element) if and only if xⁿ = e for any positive integer n.

Proof of (⇒):

Assume that

\((y^(-1)xy)^n = e.\)

We need to prove that xⁿ = e.

4. Expanding (y⁻¹xy)ⁿ, we have

\((y^(-1)xy)(y^(-1)xy)...(y^(-1)xy) = e,\)

where there are n terms.

5. By associativity, we can rearrange the expression as

\((y^(-1))(x(y(y^(-1))))(xy)...(y^(-1)xy) = e.\)

6. Since

\((y^(-1))(y) = e\)

(the inverse of y times y is the identity element), we can simplify the expression to

\((y^(-1))(xy)(y(y^(-1))))(xy)...(y^(-1)xy) = e.\)

7. By canceling adjacent inverses, we get

\((y^(-1))(xy)(xy)...(y^(-1)xy) = e.\)

8. Further simplifying, we have

\((y^(-1))(x(xy)...(y^(-1)xy)) = e.\)

9. Since y⁻¹ and y⁻¹xy are both elements of the group G, their product must also be in G.

10. Therefore, we have

\((y^(-1))(x(xy)...(y^(-1)xy)) = e \: implies \: x^n = e, where \: n = (o(y^(-1)xy)).\)

Proof of (⇐):

Assume that xⁿ = e. We need to prove that (y⁻¹xy)ⁿ = e.

11. From xⁿ = e, we can rewrite it as xⁿ =

\(x^o(x) = e.\)

(Since the order of an element x is defined as the smallest positive integer n such that xⁿ = e.)

12. Multiplying both sides by y⁻¹ from the left, we have (y⁻¹)xⁿ = (y⁻¹)e.

13. By associativity, we can rearrange the expression as (y⁻¹x)ⁿ = (y⁻¹)e.

14. Since (y⁻¹)e = y⁻¹ (the inverse of the identity element is itself), we get (y⁻¹x)ⁿ = y⁻¹.

15. Multiplying both sides by y from the left, we have y(y⁻¹x)ⁿ = yy⁻¹.

16. By associativity, we can rearrange the expression as (yy⁻¹)(y⁻¹x)ⁿ = yy⁻¹.

17. Since (yy⁻¹) = e, we get e(y⁻¹x)ⁿ = e.

18. By the definition of the identity element, e(x)ⁿ = e.

19. Since eⁿ = e, we have (x)ⁿ = e.

20. By the definition of the order of x, we conclude that o(x) divides n, i.e., o(x) | n.

21. Let n = o(x) * m for some positive integer m.

22. Substituting this into (y⁻¹x)ⁿ = y⁻¹, we get

\((y^(-1)x)^(o(x) * m) = y^(-1).\)

23. By the property of exponents, we have

\([(y^(-1)x)^(o(x))]^m = y^(-1).\)

24. Since

\([(y^(-1)x)^(o(x))]\)

is an element of G, its inverse must also be in G.

25. Taking the inverse of both sides, we have

\([(y^(-1)x)^(o(x))]^(-1)^m = (y^(-1))^(-1).\)

26. Simplifying the expression, we get

\([(y^(-1)x)^(o(x))]^m = y.\)

27. Since

\((y^(-1)x)^(o(x)) = e\)

28. Since eᵐ = e, we conclude that y = e.

29. Therefore,

\((y^(-1)xy)^n = (e^(-1)xe)^n = x^n = e.\)

From steps 4 to 29, we have shown both (⇒) and (⇐), which proves that o(x) = o(y⁻¹xy) for all x, y in every group G.

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The five elements to be assessed during sample selection in audit sampling are Sapmlinf Frame, Sample Size, Sampling Method, Sampling Interval, Sampling Risk.

1. Sampling Frame: The sampling frame is the list or source from which the sample will be selected. It is important to ensure that the sampling frame represents the entire population accurately and includes all relevant elements.

2. Sample Size: Determining the appropriate sample size is crucial to ensure the sample is representative of the population and provides sufficient evidence for drawing conclusions. Factors such as desired confidence level, acceptable level of risk, and variability within the population influence the determination of the sample size.

3. Sampling Method: There are various sampling methods available, including random sampling, stratified sampling, and systematic sampling. The chosen sampling method should be appropriate for the objectives of the audit and the characteristics of the population.

4. Sampling Interval: In certain sampling methods, such as systematic sampling, a sampling interval is used to select elements from the population. The sampling interval is determined by dividing the population size by the desired sample size and helps ensure randomization in the selection process.

5. Sampling Risk: Sampling risk refers to the risk that the conclusions drawn from the sample may not be representative of the entire population. It is important to assess and control sampling risk by considering factors such as the desired level of confidence, allowable risk of incorrect conclusions, and the precision required in the audit results.

During the sample selection process, auditors need to carefully consider these elements to ensure that the selected sample accurately represents the population and provides reliable results. By assessing and addressing these elements, auditors can enhance the effectiveness and efficiency of the audit sampling process, allowing for meaningful generalizations about the population.

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Answer:

-3y (y^2+ 2y-2)

Step-by-step explanation:

i took the test and got this answer

Answer:

107.5 cm

Step-by-step explanation:

find the length of the missing side of triangle which is 22.5 rounded then add all the sides together

Step-by-step explanation:

So a dice has 6 faces, and one of those faces is 4, so the chance of rolling a 4 is 1 in 6 chance, 1/6. Since the five others faces are not 4, the chance of not rolling a 4 is 5/6.

The number of mangoes = 48

The number of grapefruit = 12

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;

A box of fruit has four times as many oranges as grapefruit.

And, Total number of fruit = 60 pieces

Now,

Let number of grapefruit = x

Then, Number of Mangoes = 4x

So, We can formulate;

⇒ x + 4x = 60

Solve for x as;

⇒ 5x = 60

⇒ x = 12

Thus, Number of grapefruit = x

                                          = 12

Then, Number of Mangoes = 4x

                                          = 4 × 12

                                          = 48

Therefore, The number of mangoes = 48

The number of grapefruit = 12

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Answer:

its divided into sixths

Based on the information that has been provided, it is not possible to conclude whether there are a greater number of philosophers or mathematicians.

The statement only provides information regarding the proportion of mathematicians to philosophers and vice versa; it does not provide information regarding the total number of people working in either subject. It is possible that there are a greater number of philosophers, a greater number of mathematicians, or an equal number of both. We would need further information, such as the overall number of people in the fields, in order to accurately establish the number of persons who are present in each field specifically.

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If both p and 2p+1 are prime, then 2p+1 is a safe prime and p is a Sophie Germain prime.

Sophie Germain was a French mathematician who used these primes to investigate Fermat's Last Theorem, a famous mathematical conjecture that was finally proved in 1994 by Andrew Wiles. A safe prime is a prime number of the form 2p+1, where p is also a prime number, and it is called "safe" because it has some cryptographic properties that make it useful in certain encryption schemes.                                                                                                                                          In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.

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The number of ways to arrange the books when one shelf is left empty is m * (m-1)^(n-1).To solve this problem, we need to use the permutation formula since the order of books on shelves matters.

We can start by finding the total number of ways to arrange n books on m shelves, where n is the number of books and m is the number of shelves.

If we have n books and m shelves, the number of ways to arrange the books on the shelves without any restrictions is m^n. However, since we have the additional restriction that each shelf must contain at least one book, we need to subtract the number of arrangements where one or more shelves are left empty.

The number of ways to arrange the books when one shelf is left empty is m * (m-1)^(n-1), since we can choose the empty shelf in m ways, and then arrange the remaining n-1 books on the (m-1) non-empty shelves in (m-1)^(n-1) ways. However, we need to be careful not to double-count the arrangements where two shelves are left empty, so we add back in the number of arrangements where two shelves are empty, which is m choose 2 * (m-2)^(n-1). We continue this process until we've accounted for all the possible combinations of empty shelves.

Therefore, the total number of arrangements with at least one book on each shelf is:

m^n - m*(m-1)^(n-1) + m choose 2 * (m-2)^(n-1) - m choose 3 * (m-3)^(n-1) + ...

This expression can be simplified using the principle of inclusion-exclusion, but it still depends on the values of n and m.

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Horizontal shift add 4 inside the parenthesis: f(x + 4) = (x+4)^2

Add the vertical shrink factor to the outside of the function: (1/2)f(x)

Combined f(x) = (1/2)(x+4)^2

the qestion is incomplete .please read below to find the missing content

Write the equation for the following function that represents a horizontal shift left 4 and a vertical shrink by a factor of ½ on the parent function f(x) = x ^ 2

A horizontal line is a line that runs from left to right. When you watch the sunrise over the horizon, you see the sun rise above the horizon. The x-axis is an example of a horizontal line.

Horizontal lines are straight lines mapped from left to right and vertical lines are straight lines mapped from top to bottom. In coordinate geometry, lines parallel to the x-axis are called horizontal lines, and lines parallel to the y-axis are called vertical lines.

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Answer:

Equal volumes

Step-by-step explanation:

Given

\(h \to\) height of cone

\(d \to\) diameter of cone

\(H \to\) height of cylinder

\(D \to\) diameter of cylinder

Such that:

\(d = 2D\)

\(h =\frac{3}{4}H\)

Required

The relationship between the volumes

The volume of a cylinder is:

\(V_1 = \pi R^2H\)

Where

\(R = 0.5D\)

So:

\(V_1 = \pi (0.5D)^2H\)

\(V_1 = \pi *0.25*D^2H\)

\(V_1 = 0.25\pi D^2H\)

The volume of the cone is:

\(V_2 = \frac{1}{3}\pi r^2h\)

Where

\(r =0.5d\)

\(r = 0.5 * 2D\)

\(r = D\)

and

\(h =\frac{3}{4}H\)

So, we have:

\(V_2 = \frac{1}{3}\pi * D^2 * \frac{3}{4}H\)

\(V_2 = \frac{1}{4}\pi * D^2H\)

\(V_2 = 0.25\pi * D^2H\)

\(V_2 = 0.25\pi D^2H\)

So, we have:

\(V_1 = 0.25\pi D^2H\)

\(V_2 = 0.25\pi * D^2H\)

\(V_1 = V_2\)

Answer: : When a cone and cylinder have the same height and radius the cone will fit inside the cylinder. The volume of the cone will be one-third that of the cylinder. If the radius or height are different, then there is no relationship between them.

Step-by-step explanation:

Using binomial distribution, the probability of

a. 5 will still be with the company after 1 year is 28%.

b. at most 6 will still be with the company after 1 year is 89%.

The probability of a new employee in a fast-food chain still being with the company at the end of the year is given to be 0.6, which can be taken as the success of the experiment, p.

We are finding probability for people, thus, our sample size, n = 8.

Thus, we can show the given experiment a binomial distribution, with n = 8, and p = 0.6.

(i) We are asked for the probability that 5 will still be with the company.

Thus, we take x = 5.

P(X = 5) = (8C5)(0.6⁵)((1 - 0.6)⁸⁻⁵),

or, P(X = 5) = (56)(0.07776)(0.064),

or, P(X = 5) = 0.27869184 ≈ 0.28 or 28%.

(ii) We are asked for the probability that at most 6 will still be with the company.

Thus, our x = 6, and we need to take all values below it also.

P(X ≤ 6)

= 1 - P(X > 6)

= 1 - P(X = 7) - P(X = 8)

= 1 - (8C7)(0.6)⁷((1 - 0.6)⁸⁻⁷) - (8C8)(0.6)⁸((1 - 0.6)⁸⁻⁸)

= 1 - 8*0.0279936*0.4 - 1*0.01679616*1

= 1 - 0.08957952 - 0.01679616

= 0.89362432 ≈ 0.89 or 89%.

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The provided question is incomplete. The complete question is:

If the probability of new employee in a fast-food chain still being with the company at the end of 1 year is 0.6, what is the probability that out of 8 newly hired people,

a. 5 will still be with the company after 1 year?

b. at most 6 will still be with the company after 1 year?

Answer:

the answer is  {Corelli, Yao}

Step-by-step explanation:

i took the quiz

the answer is  {Corelli, Yao}

it them bc of they why of the number is place and it repeied

Answer:

Hello! The answer to your question is (-7, -4).

Step-by-step explanation:

To solve this problem, you have to use the midpoint formula:

\((\frac{x1 + x2}{2} , \frac{y1 + y2}{2} )\)

Now, we can substitute our values in:

\((\frac{-9 + -5}{2}, \frac{1 - 9}{2} )\\= (\frac{-14}{2} , \frac{-8}{2}) \\= (-7, -4)\)

The total area under the piecewise function on the interval [0, 6], sum the areas of the triangle and the semi-circle:
Total area = Area of triangle + Area of the semi-circle
Total area = 2 + 2π square units

To find the area under the given piecewise function on the interval [0, 6], we can break the problem into two parts based on the two given functions:
1. f(x) = x, 0 < x < 2
2. f(x) = √(4 - (x - 4)²) + 2, 2 < x < 6

First, consider the function f(x) = x on the interval [0, 2]. The graph of this function is a straight line with a slope of 1. The area under this function forms a triangle with a base of length 2 and a height of 2. The area of this triangle can be found using the formula for the area of a triangle:
Area = (1/2) × base × height
Area = (1/2) × 2 × 2
Area = 2 square units

Now, consider the function f(x) = √(4 - (x - 4)²) + 2 on the interval [2, 6]. This function describes a semi-circle with a radius of 2 centered at the point (4, 2). The area of a semi-circle can be found using the formula for the area of a circle:
Area of semi-circle = (1/2) × π × radius²
Area of semi-circle = (1/2) × π × 2²
Area of semi-circle = 2π square units
Finally, to find the total area under the piecewise function on the interval [0, 6], sum the areas of the triangle and the semi-circle:
Total area = Area of triangle + Area of the semi-circle
Total area = 2 + 2π square units

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The system of inequalities used to determine the total number of hot dogs and pretzels Ben can sell is:n 0.50h + 0.40p ≤ 100

h + p ≥ 200

To determine the total number of hot dogs and pretzels Ben can sell, we need to establish an inequality based on the given conditions. Let's analyze the given information:

The cost of each hot dog is $0.50, and the cost of each pretzel is $0.40.

Ben has only $100 to spend each day on hot dogs and pretzels.

Ben wants to sell at least 200 items each day.

Let h represent the number of hot dogs and p represent the number of pretzels that Ben sells.

The cost of h hot dogs would be 0.50h, and the cost of p pretzels would be 0.40p. According to the given condition, Ben has only $100 to spend each day on hot dogs and pretzels. So, we can write the following inequality:

0.50h + 0.40p ≤ 100

This inequality ensures that the total cost of the hot dogs and pretzels sold does not exceed Ben's budget of $100.

Additionally, Ben wants to sell at least 200 items each day. The total number of items sold is the sum of the number of hot dogs and pretzels, which is h + p. So, we can write another inequality:

h + p ≥ 200

This inequality ensures that the total number of items sold is at least 200.

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The solution is Option C.

The L'Hopital's rule is applied to the equation lim x approaches 0 (1-cosx)/x

L'Hopital's rule then states that the slope of the curve when t = c is the limit of the slope of the tangent to the curve as the curve approaches the origin, provided that this is defined. The limit of a quotient of functions (i.e., an algebraic fraction) is equal to the limit of their derivatives.The tangent to the curve at the point [g(t), f(t)] is given by [g′(t), f′(t)]

And , lim x approches c  [ f ( x ) / g ( x ) ] = lim x approches c [ f' ( x ) / g' ( x ) ]

Given data ,

Let the equation be represented as A

Now , the value of A is

a)

The equation is A = lim x approaches 0 (1/x)

On simplifying the equation , we get

The limit diverges as the function diverges and limit does not exist

And ,  lim x approaches 0₊ (1/x) ≠ lim x approaches 0₋ (1/x) = ∞

b)

The equation is A = lim x approaches 0 ( 2x² - 1 ) / ( 3x - 1 )

On simplifying the equation , we get

when x = 0 ,

Substitute the value of x = 0 in the limit , we get

A = ( 2 ( 0 )² - 1 ) / ( 3 ( 0 ) - 1 )

A = ( 0 - 1 ) / ( 0 - 1 )

A = 1

c)

The equation is A = lim x approaches 0 ( 1 - cosx ) / x

On simplifying the equation , we get

Applying L'Hopital's rule , we get

lim x approches c  [ f ( x ) / g ( x ) ] = lim x approches c [ f' ( x ) / g' ( x ) ]

f ( x ) = ( 1 - cos x )

g ( x ) = x

f' ( x ) = sin x

g' ( x ) = 1

So ,

lim x approches 0 [ f' ( x ) / g' ( x ) ] = lim x approches 0 ( sin x / 1 )

when x = 0

sin ( 0 ) = 0

Therefore , the value of lim x approaches 0 (1-cosx)/x = 0

d)

The equation is A = lim x approaches 0 ( cos 2x ) / 2

On simplifying the equation , we get

when x = 0 ,

A = cos ( 2 ( 0 ) / 2

A = cos ( 0 ) / 2

A = 1/2

Hence , the L'Hopital's rule is applied to lim x approaches 0 ( 1 - cosx ) / x

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Using proportions, it is found that the rate of water flow was of 6 miles per hour.

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

In this problem, the bottle flowed 1 mile in 10 minutes. How much does it flow in 1 hour = 60 minutes? The rule of three is given by:

1 mile - 10 minutes

x miles - 60 minutes

Applying cross multiplication:

10x = 60

x = 60/10

x = 6.

Hence, the rate of water flow was of 6 miles per hour.

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Answer:

-Delia paid $85 in interest.

-The simple interest rate charged by the credit union is 0.85%.

Step-by-step explanation:

To find the interest she paid, first you have to determine the total payment she made and subtract the principal from that amount:

$108.50*10=$1,085

$1,085-$1,000=$85

According to this, Delia paid $85 in interest.

Now, you have to use the following formula to find the simple interest rate charged by the credit union:

r=(1/t)*(A/P-1), where:

r=rate of interest

t=time period=10

A=accrued amount=1,085

P=principal amount=1,000

r=(1/10)*(1,085/1,000-1)

r=0.1*0.085

r=0.0085→0.85%

According to this, the simple interest rate charged by the credit union is 0.85%.

Step-by-step explanation:

To find the smallest total number of hot dogs that Yadira's mom can purchase, we need to find the least common multiple (LCM) of 12 and 9, since she wants the same number of hot dogs as hot dog buns.

The multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...

The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ...

The least common multiple of 12 and 9 is 36, since it is the smallest number that appears in both lists. Therefore, Yadira's mom should purchase 36 hot dogs (3 packs of 12) and 36 hot dog buns (4 packs of 9) for the family barbecue.

Answer:

tried my best 2 help

Step-by-step explanation:

2.circumference = pi x diameter

c = pi x (8x2)

c= 50.2654824574

3. when  measuring physical objects,  

4. to find the length around an object such as your waist when buying a top ???? couldnt think of anything else

Answer:

12.5% decrease

Step-by-step explanation:

[(V2 - V1)  /  V1] x 100 = Percent Change

The investment earned a rate of interest of approximately 22.7%, indicating a positive growth of the investment over the two-year period.

The rate of interest earned can be calculated using the formula for compound interest:

Rate of Interest = [(Final Value - Initial Value) / Initial Value] * 100

Using the given information:

Initial Value = $3,300

Final Value = $4,050

Substituting these values into the formula:

Rate of Interest = [($4,050 - $3,300) / $3,300] * 100

Calculating the numerator:

$4,050 - $3,300 = $750

Substituting the numerator into the formula:

Rate of Interest = [$750 / $3,300] * 100

Simplifying the expression:

Rate of Interest = (750 / 3300) * 100

Dividing the numerator by the denominator:

Rate of Interest ≈ 0.227 * 100

Rate of Interest ≈ 22.7%

Therefore, the rate of interest earned on the investment is approximately 22.7%. The formula for calculating the rate of interest is derived from the concept of compound interest. It measures the percentage increase in the initial investment that results in the final value.

To calculate the interest earned, subtract the initial value from the final value. In this case, the initial value is $3,300 and the final value is $4,050, resulting in an increase of $750. Divide this increase by the initial value to get the fraction gained, then multiply by 100 to express the rate of interest as a percentage.

In the given scenario, the investment earned a rate of interest of approximately 22.7%, indicating a positive growth of the investment over the two-year period.

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Answer:

10.25

Step-by-step explanation:

205 × 5 = 1025

2.05 has 2 decimal places

5 has 0 decimal places

Therefore, the answer has 2 decimal places

1025

10.25

Answer:

ln(x) = 1/x... It's a basic rule of Calculus.

(a) The possible rational roots of p(x) are: ±1, ±2, ±4, ±8, and ±16.

(b) p(x) does not have any rational roots.

(c) The graph of p(x) will have critical points at x = -2 and x = 2.

(d) p(x) > 0 for -2 < x < 2.

(a) To find the possible rational roots of the polynomial p(x) = x³ − 12x + 16, we can use the rational root theorem. According to the theorem, the rational roots are given by the factors of the constant term (16) divided by the factors of the leading coefficient (1).

The factors of 16 are ±1, ±2, ±4, ±8, and ±16.

The factors of 1 are ±1.

Combining these factors, the possible rational roots of p(x) are:

±1, ±2, ±4, ±8, and ±16.

(b) To factor p(x) completely into linear factors, we need to find its roots. We can use synthetic division or other root-finding methods to determine the roots. Let's check the possible rational roots we found in part (a) to see which ones are actual roots of p(x).

Let's start with x = 1:

   1 |  1  0  -12   16

      |     1    1 -11

    -----------------

      |  1  1   -11   5

Since the remainder is not zero, x = 1 is not a root of p(x).

Next, let's try x = -1:

  -1 |  1  0  -12   16

      |    -1    1  11

    -----------------

      |  1 -1   -11  27

Again, the remainder is not zero, so x = -1 is not a root of p(x).

Continuing this process, we find that none of the possible rational roots ±1, ±2, ±4, ±8, and ±16 are roots of p(x). Therefore, p(x) does not have any rational roots.

(c) Since p(x) does not have rational roots, we cannot factor it into linear factors.

(d) To determine when p(x) > 0, we need to examine the intervals where the graph of p(x) is above the x-axis.

To sketch a graph of p(x), we can start by finding the critical points. We can do this by finding where the derivative of p(x) is equal to zero.

Taking the derivative of p(x) with respect to x, we get:

p'(x) = 3x² - 12

Setting p'(x) = 0 and solving for x:

3x² - 12 = 0

3x² = 12

x² = 4

x = ±2

The critical points of p(x) are x = -2 and x = 2.

Now, let's consider the behavior of p(x) around these critical points.

When x < -2:

p(x) = x³ - 12x + 16 < 0

When -2 < x < 2:

p(x) = x³ - 12x + 16 > 0

When x > 2:

p(x) = x³ - 12x + 16 > 0

Therefore, p(x) is greater than zero for -2 < x < 2.

To summarize:

(a) Possible rational roots of p(x): ±1, ±2, ±4, ±8, ±16.

(b) p(x) cannot be factored into linear factors.

(c) The graph of p(x) will have critical points at x = -2 and x = 2.

(d) p(x) > 0 for -2 < x < 2.

Learn more about rational roots here:

https://brainly.com/question/29551180

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