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The derivative of f(x) = (1 + x4 - 1/x)5/3 using the chain rule is (5/3)(1 + x4 - 1/x)2/3(4x3 + 1/x2).

To find the derivative of f(x) = (1 + x^4 - 1/x)^5/3 using the chain rule, we need to use the following steps:

1. Identify the inner function and the outer function. In this case, the inner function is 1 + x^4 - 1/x and the outer function is ( )^5/3.
2. Find the derivative of the outer function with respect to the inner function. This is done by using the power rule: (5/3)( )^2/3.
3. Find the derivative of the inner function with respect to x. This is done by using the power rule and the quotient rule: 4x^3 + 1/x^2.
4. Multiply the derivative of the outer function and the derivative of the inner function together to get the derivative of the original function: (5/3)(1 + x^4 - 1/x)^2/3(4x^3 + 1/x^2).

Therefore, the derivative of f(x) = (1 + x^4 - 1/x)^5/3 using the chain rule is (5/3)(1 + x^4 - 1/x)^2/3(4x^3 + 1/x^2).

Here is the answer formatted in HTML:

To find the derivative of f(x) = (1 + x4 - 1/x)5/3 using the chain rule, we need to use the following steps:

Therefore, the derivative of f(x) = (1 + x4 - 1/x)5/3 using the chain rule is (5/3)(1 + x4 - 1/x)2/3(4x3 + 1/x2).

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The total amount of interest on the investment if compounded continuously comes out to be $10064.60, rounded off to the nearest dollar and cent.

To find the total amount of money accumulated for an initial investment of $5200 at 6% compounded quarterly after 11 years, we can use the formula A = P(1+r/n)^(nt), where A is the total amount, P is the principal or initial investment, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Plugging in the given values, we get:

A = 5200(1+0.06/4)^(4*11)
A = 5200(1.015)^44
A = 5200*1.9878
A = $10336.96

Therefore, the total amount of money accumulated after 11 years is $10336.96, rounded off to the nearest dollar and cent.

To compute the total amount of interest on the investment if compounded continuously, we can use the formula A= P e^(rt), where e is the base of the natural logarithm. Plugging in the given values, we get:

A = 5200 e^(0.06*11)
A = 5200 e^0.66
A = 5200*1.9355
A = $10064.60

Therefore, the total amount of interest on the investment if compounded continuously is $10064.60, rounded off to the nearest dollar and cent.

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Answer:

not a function

Step-by-step explanation:

Answer:

not a function

Step-by-step explanation:

The number that must be added to both sides of the equation x2 + 12x = 13 so that the method of completing the square can be used to solve the roots is 36. The correct answer is C. 36.

To complete the square, we need to add the square of half the coefficient of the x term to both sides of the equation. In this case, the coefficient of the x term is 12, so half of it is 6. The square of 6 is 36, so we need to add 36 to both sides of the equation. Hence the correct option is C) 36.

The equation then becomes:
x2 + 12x + 36 = 13 + 36

Simplifying the right side of the equation gives us:
x2 + 12x + 36 = 49

Now we can factor the left side of the equation into a perfect square:
(x + 6)2 = 49

From here, we can use the square root property to solve for x:
x + 6 = ±√49

Solving for x gives us:
x = -6 ± 7

So the roots of the equation are x = 1 and x = -13.

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This is the appropriate set notation for the set J, which includes all integers between -5 and -3.

The set J can be rewritten by listing its elements in the appropriate set notation. Since the set J contains all integers between -5 and -3, we can list the elements as follows:

An integer is the number zero, a positive natural number or a negative integer with a minus sign. The negative numbers are the additive inverses of the corresponding positive numbers.

J = {-5, -4}

In set notation, this can be written as:

J = {x | x is an integer and -5 <= x < -3}

Therefore, the set J can be rewritten as:

J = {-5, -4}

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The approximate solution using the three-point formula with step size h = 1/2 is y(t) = 0 for all t.

The boundary value problem given is y′′ − y = 0 with boundary conditions y(0) = 0 and y(2) = e2 − e−2.

(a) To find the exact solution y(t), we can use the characteristic equation r^2 - 1 = 0. This gives us r = 1 and r = -1. The general solution is therefore y(t) = c1e^t + c2e^-t.

Using the boundary conditions, we can find the constants c1 and c2.

For y(0) = 0, we have 0 = c1 + c2, which gives us c2 = -c1.

For y(2) = e2 − e−2, we have e2 − e−2 = c1e^2 + c2e^-2. Substituting c2 = -c1, we get e2 − e−2 = c1e^2 - c1e^-2.

Solving for c1, we get c1 = (e2 − e−2)/(e^2 - e^-2) = 1/2. Therefore, c2 = -1/2.

The exact solution is y(t) = (1/2)e^t - (1/2)e^-t.

(b) To use the three-point formula with step size h = 1/2, we can set up a table with tn and yn values.

tn | yn
---|---
0  | 0
1/2| y1
1  | y2
3/2| y3
2  | e2 - e-2

The three-point formula is yn+1 = yn-1 + 2h(y′n). We can use this formula to find the values of y1, y2, and y3.

For y1, we have y1 = 0 + 2(1/2)(y′0) = y′0. Since y′0 = y′(0) = (1/2)e^0 - (1/2)e^0 = 0, we have y1 = 0.

For y2, we have y2 = y0 + 2(1/2)(y′1) = 0 + 2(1/2)(0) = 0.

For y3, we have y3 = y1 + 2(1/2)(y′2) = 0 + 2(1/2)(0) = 0.

Therefore, the approximate solution using the three-point formula with step size h = 1/2 is y(t) = 0 for all t.

It is important to note that the three-point formula is not accurate for this particular boundary value problem due to the size of the step and the nature of the differential equation. A smaller step size or a different numerical method may yield a more accurate approximation.

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Simplifying x^7/x^3 we get x^4.

We have the numerator of the given fraction x^7/x^3 as x^7, that is x raised to the power 7. We can rewrite x^7 equals to (x^4)(x^3) , that is x raised to the power 4 multiplied by x raised to the power 3 (by exponential method).

Similary, the denominator of the given fraction x^7/x^3 is x^3, that is x raised to the power 3.

Rewritting the fraction x^7/x^3 as,

(x^4)(x^3) / x^3  { using division rule}

= x^4  (that is, x raised to the power 4)

Thus after simplifying x^7/x^3 (using division rule) we get x^4.

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Just multiplying the amount of possibilities in each category together will give us the total number of meals that can be made using the options provided.

This is known as the Fundamental Counting Principle. Using this principle, the total number of possible meals is:

6 main courses × 4 vegetables × 2 salads × 3 beverages = 144 possible meals

Therefore, there are 144 possible meals that can be made by choosing from the given options.

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As seen in the attached picture, the account by Grant and the painting are consistent with each other since they both depict a serious and formal situation.

Ulysses S. Grant was an American general and statesman who served as the 18th President of the United States. He was born in 1822 in Ohio and graduated from the United States Military Academy at West Point in 1843. He served in the Mexican-American War and later became a leading Union general in the American Civil War.

As we compare his account of Lee's rendition and the painting provided, we can conclude that they are indeed consistent. Both show how serious and formal the meeting was. Grant even describes the way Lee was dressed, which is confirmed by the painting.

The picture and the account appear in the attachment below.

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Answer:

-5 - 2sqrt6

Step-by-step explanation:

To simplify (sqrt2+sqrt3)/(sqrt2-sqrt3), we can rationalize the denominator, which involves multiplying both the numerator and denominator by the conjugate of the denominator.

The conjugate of sqrt2-sqrt3 is sqrt2+sqrt3, so we can multiply both the numerator and denominator by sqrt2+sqrt3:

(sqrt2+sqrt3)/(sqrt2-sqrt3) * (sqrt2+sqrt3)/(sqrt2+sqrt3) = ((sqrt2+sqrt3)(sqrt2+sqrt3))/((sqrt2-sqrt3)(sqrt2+sqrt3))

Expanding the numerator and simplifying, we get:

(sqrt2+sqrt3)^2 / (sqrt2^2 - sqrt3^2)

= (2 + 2sqrt2sqrt3 + 3) / (2 - 3)

= (5 + 2sqrt6) / (-1)

= -5 - 2sqrt6

Therefore, (sqrt2+sqrt3)/(sqrt2-sqrt3) simplifies to -5 - 2sqrt6.

The simplest form of the expression (√2 + √3) / (√2 - √3) is - (5 + 2√6).

We have,

To simplify the expression (√2 + √3) / (√2 - √3), we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of (√2 - √3) is (√2 + √3).

Multiplying the numerator and denominator by (√2 + √3), we get:

[(√2 + √3) x (√2 + √3)] / [(√2 - √3) x (√2 + √3)]

Expanding both the numerator and denominator:

[(√2 + √3)(√2 + √3)] / [√2 x √2 - √2 x √3 + √3 x √2 - √3 x √3]

Simplifying:

[2 + 2√2√3 + 3] / [2 - 3]

Combining like terms:

[5 + 2√6] / [-1]

Since the denominator is -1, we can multiply both the numerator and denominator by -1 to simplify further:

[5 + 2√6] / 1

Finally, we have:

(5 + 2√6)

Therefore,

The simplest form of the expression (√2 + √3) / (√2 - √3) is - (5 + 2√6).

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The amount of cost for the fabric is given by A = $ 13

Equations are statements in mathematics that have two algebraic expressions on either side of the equals (=) sign.

It displays the similarity of the connections between the phrases on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are examples of the parts of an equation. When creating an equation, the "=" symbol and terms on both sides are necessary.

Given data ,

Let the total cost of red fabric be represented as A

Now , the amount of red fabric be = 6 1/2 yards

And , the cost of red fabric per yard = $ 2

So , the total cost of red fabric A = amount of red fabric x cost of red fabric per yard

On simplifying the equation , we get

The total cost of red fabric A = ( 6 1/2 ) x 2

The total cost of red fabric A = ( 13/2 ) x 2

The total cost of red fabric A = $ 13

Therefore , the value of A is $ 13

Hence , the total cost of red fabric is $ 13

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The correct answer is B. 25/3.

We can use the equation of a parabola with a focus at (h, k) and a directrix at y = k + p to find the depth of the reflector. The equation is:
(y - k)² = 4p(x - h)

Since the focus is at the light center, we can set h = 0 and k = 0. The distance of the focus from the vertex is 3/4 cm, so p = 3/4. The diameter of the reflector is 10 cm, so the x-coordinate of the vertex is 5 cm. We can plug in these values to find the depth of the reflector:
(y - 0)² = 4(3/4)(x - 0)
y² = 3x
y = √(3x)

When x = 5, we can find the depth of the reflector:

y = √(3*5)
y = √15
y = 3.87 cm
The depth of the reflector is 3.87 cm, or 25/3 cm.

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When a car headlight reflector is cut by a plane along its axis, the section obtained is a parabola. This parabola is such that the light center is at the focus. The distance of focus from the vertex is 3/4 cm and the diameter of the reflector is 10 cm. The depth of the reflector comes out to be CD = VC - VF = (10/3) - (3/4) = 27/4 cm

The vertex of the parabola is the midpoint of the diameter of the reflector. Let V be the vertex of the parabola and let F be the focus. The distance between V and F is given as 3/4 cm.The reflector is such that light rays from the source (headlamp) placed at the focus of the parabola are reflected by the parabola in such a way that the rays are parallel to the axis of the parabola. This is known as the reflecting property of the parabola.

This is equal to CD.Let P be the point on the parabola, as shown in the diagram below, such that PF is equal to the diameter of the reflector. Then, by the definition of the parabola, the distance from P to the vertex C is the same as the distance from the focus F to P, i.e., PF = PC. Since PF is equal to the diameter of the reflector, it is given that PF = 10 cm.Therefore, PC = 10 cm. It is also given that VF = 3/4 cm. Therefore, VC = PC - PV = 10 - 20/3 = 10/3 cm.

Hence, the depth of the reflector is CD = VC - VF = (10/3) - (3/4) = 27/4 cm. Therefore, the depth of the reflector is 27/4 cm, which is the correct option among the given choices.

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The simplified expression is  [tex]8x^4y^4[/tex].

Simplifying an expression is just another way to say solving a math problem. When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do. For example, take this expression: 4 + 6 + 5.

Here the given expression is

=> [tex](4x^3y^3)(2x.2y)[/tex]

=> [tex]4\times x^3\times y^3\times2x\times2y[/tex]

=> [tex]4\times2\times2\times x^3\times x \times y^3\times y[/tex]

=> 8[tex]\times x^{3+1}\times y^{3+1}[/tex]

=> [tex]8x^4y^4[/tex]

Hence the simplified expression is  [tex]8x^4y^4[/tex].

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Answer:

Trust me I guarantee you it's correct.

Step-by-step explanation:

D. 8x^5y^4

The requried value of the first quartile is 4.5.

Interquartile range (IQR): The IQR is the range of the middle 50% of values in a data set. To calculate the IQR, we first need to find the quartiles of the data set.

To find the first quartile (Q1), we need to arrange the given values in ascending order and then find the median of the lower half of the values.

The given values arranged in ascending order are:

3, 6, 8, 11

The lower half of the values are:

3, 6

The median of the lower half is:

(Q1) = (3 + 6)/2 = 4.5

Therefore, the value of the first quartile is 4.5.

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The simplified root of the given root i.e. √(400x¹⁰⁰) is 20x⁵⁰. The solution has been obtained by using the law of indices.

What is the law of indices?

The guidelines for simplifying expressions containing powers of the same base number are known as index laws.

We are given an expression as √(400x¹⁰⁰).

Using law of indices,

⇒ √(400x¹⁰⁰) = √400 * √(x¹⁰⁰)             ( as √ab = √a * √b)

⇒ √(400x¹⁰⁰) = 20 √(x¹⁰⁰)

⇒ √(400x¹⁰⁰) = 20 √(x⁵⁰ * x⁵⁰)             ( as a⁴ = a² + a²)

⇒ √(400x¹⁰⁰) = 20x⁵⁰

Hence, the simplified root of the given root i.e. √(400x¹⁰⁰) is 20x⁵⁰.

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To calculate the probability that a randomly selected person has at least 2 TV's, you would first calculate the probability of a person having more than 2 TV's:

Then, you would calculate the probability of a person having less than 2 TV's:

Finally, you would calculate the probability of a person having exactly 2 TV's:

Therefore, the probability that a randomly selected person has at least 2 TV's is:

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The simplest form is (x+4)/(x-5). Restrictions on the variable are that x ≠ 5  and x≠-4.

The given expression is (x^(2)+8x+16)/(x^(2)-x-20). In order to simplify this expression, we need to factor the numerator and denominator and then cancel out any common factors.

The numerator can be factored as (x+4)(x+4) and the denominator can be factored as (x-5)(x+4).

So the expression becomes:

(x+4)(x+4)/(x-5)(x+4)

Now we can cancel out the common factor of (x+4):

(x+4)/(x-5)

Therefore, the simplest form of the expression is (x+4)/(x-5).

The restrictions on the variable are that x cannot be equal to 5 or -4, because these values would make the denominator equal to zero and the expression would be undefined.

So the final answer is (x+4)/(x-5) with restrictions x≠5 and x≠-4.

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number 8 (i think) is d, 1 in 1,500, and number 9 is c, 1,513 cars :)

The graphs of y = x and y = -1 are related to each other because they intersect at the point (-1, -1)

An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.

The standard equation of a linear equation is:

y = mx + b

where m is the rate of change and b is the y intercept

Given the graphs of y = x and y = -1.

As we can see from the graphs, they intersect at the point (-1, -1)

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As a result, the 15 cartons of creamer will last for 1,350 days, or nearly 3.7  expression years. This implies the consumer drinks one serving of creamer per day at a weight of 5 grammes per serving.

An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematics, and form. They are employed in the depiction of mathematical formulas, the solving of equations, and the simplification of mathematical relationships.

(Total amount of creamer in grammes) / number of days (Amount of creamer consumed per day in grams)

Let's start by calculating the entire amount of creamer in grammes:

(Number of cartons) x (Total quantity of creamer) (Amount of creamer per box)

15 boxes x 450 grammes each box = total quantity of creamer

Total creamer weight = 6,750 g

(Total amount of creamer in grammes) / number of days (Amount of creamer consumed per day in grams)

The number of days is 6,750 grammes divided by 5 grammes each day.

The number of days is 1,350.

As a result, the 15 cartons of creamer will last for 1,350 days, or nearly 3.7 years. This implies the consumer drinks one serving of creamer per day at a weight of 5 grammes per serving.

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Answer:

i need more details i cant answer.

The equation of the red graph is g(x) = x² + 3.

Option C.

The equation of the red graph can be determined by considering upscaling or transformation on the x - axis.

Since the equation focuses on the function g(x) and not on upscaling in the y-axis, any changes should not affect the x-axis. Therefore, we can eliminate (x - 3)² and (x + 3)² from the options given, since they involve modifying the x-axis.

In conclusion, as the transformation involves increasing the y-values along the y-axis, the correct choice is x² + 3 rather than x² - 3.

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The exact value of cos(5π/12), using the cosine summation identity, is (√6 - √2)/4.

The exact value of cos(5π/12) can be found using the cosine sum identity, which is:

cos(a + b) = cosa · cosb - sina · sin b

In this case, we can rewrite 5π/12 as (π/4) + (π/6) and use the identity:

cos(5π/12) = cos[(π/4) + (π/6)] = cos(π/4) · cos (π/6) - sin(π/4) · sin (π/6)

Using the values of cos(π/4) = √2/2, cos(π/6) = √3/2, sin(π/4) = √2/2, and sin(π/6) = 1/2, we can plug them into the equation:

cos(5π/12) = (√2/2) · (√3/2) - (√2/2) · (1/2) = √6/4 - √2/4 = (√6 - √2)/4

Therefore, the exact value of cos(5π/12) is (√6 - √2)/4.

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The rule for the function g(x) when completed is g(x) = -10, -15 ≤ x < -10; g(x) = -8, -10 ≤ x < -8; g(x) = -6, -8 ≤ x < -1; g(x) = 2, -1 ≤ x < 1; g(x) = 4, 1 ≤ x < 10; g(x) = 8, 10 ≤ x < 15

Given

The graph of the function g(x) such that the function g(x) is a piecewise function and each sub-function is represented by horizontal lines

To complete the function definition, we write out the y value and the domain of the functions based on the current domain

Following the above statements, we have the following function definition for g(x)

g(x) = -10, -15 ≤ x < -10

g(x) = -8, -10 ≤ x < -8

g(x) = -6, -8 ≤ x < -1

g(x) = 2, -1 ≤ x < 1

g(x) = 4, 1 ≤ x < 10

g(x) = 8, 10 ≤ x < 15

The above is the definition of the function g(x)

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Answer:

Step-by-step explanation: point form: (4,1)

x=4 , y=1

Answer:
To calculate the time at which they flash together we need to find the least common multiple of 3 and 8.

Multiples of 3 =

3 x 1 = 3​

3 x 2 = 6​

3 x 3 = 9​

3 x 4 = 12​

3 x 5 = 15​

3 x 6 = 18​

3 x 7 = 21​

3 x 8 = 24​

3 x 9 =27​

3 x 10 = 30

Multiples of 8 =

8 x 1 = 8​

8 x 2 = 16​

8 x 3 = 24

8 x 4 = 32

8 x 5 = 40​

8 x 6 = 48

8 x 7 = 56​

8 x 8 = 64​

8 x 9 =72​

8 x 10 = 80

So LCM of 3 and 8 = 24

So both the lights will flash next at 7:24

Step-by-step explanation:

-6x+25y+66z=232 is the equation of the plane and distance between the point D and the plane is  157/70.

lets find the equation of the line AB and BC and equations of the lines will be of the form

[tex]  \text{ $\frac{x-a}{l}$ = $\frac{y-b}{m}$ = $\frac{z-c}{n}$ } [/tex]

So, the equation of AB is

[tex]  \text{ $\frac{x}{2}$ = $\frac{y-4}{18}$ = $\frac{z-2}{7}$ } [/tex]

So, the equation of AB is

[tex]  \text{ $\frac{x-3}{-1}$ = $\frac{y+2}{24}$ = $\frac{z}{9}$ } [/tex]

let b1 andb2 are the direction vectors of the two above lines.

b1= 2i + 18j + 7k

b2= -1i + 24j + 9k

now n= Det( i       j       k)

                   2      18     7

                  -1       24    9

or, n= -6i + 25j + 66k

the point (0,4,2) lies on the plane. the equation of the plane passing through (0,4,2) and perpendicular to a line with a direction ratio (-6, 25, 66) is

-6x+25(y-4)+66(z-2)=0

or, -6x+25y+66z=232.

now formula of distance between the point (x0,y0,z0) and the plane is

[tex] \frac{Ax0+By0+Cz0+D}{√A^2+B^2+C^2} [/tex]

so for the point D and the plane is

( -6*0 + 25*1 + 66*2 - 132)/√5017 = 157/70.

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Answer: +-root15/2

To find the x-intercepts, set the function equal to 0 and solve for x.

0.8x^2 - 3 = 0

0.8x^2 = 3

x^2 = 15/4

x = +- root15/2

The equation x^(4)+6x^(3)-3x^(2)-24x-4=0 has possible rational roots ± 1, 2, 4, ± 1/2, 1/4.

Given the equation: $x^4+6x^3-3x^2-24x-4=0$

To identify possible rational roots we use Rational Root Theorem which states that:

If a polynomial function with integer coefficients has any rational roots then the numerator must divide the constant term and the denominator must divide the leading coefficient. Let's identify possible rational roots. The constant term is -4 and the leading coefficient is 1. Therefore, the possible rational roots are as follows:± 1, 2, 4± 1/2, 1/4

We use synthetic division to test several possible rational roots in order to identify the roots of the equation.

x−40−3−2−4−4−4−4−2+2-2+2-2+2+2-1+1-1+1-1+1+1+4-2+4-2+4-2+4+0-4+0-4+0-4±1 is the root of the equation since the remainder is zero. Therefore, divide the polynomial by x − 1.x^4+6x^3-3x^2-24x-4 = (x-1)(x^3+7x^2+4x+4x+4) = (x-1)(x^3+7x^2+8x+4)

The roots of the equation are x = 1, -2 ± i, where i = √(-1).

Hence, we have completed the following:

Possible rational roots: ± 1, 2, 4, ± 1/2, 1/4

Synthetic division to test possible rational roots: x−40−3−2−4−4−4−4−2+2-2+2-2+2+2-1+1-1+1-1+1+1+4-2+4-2+4-2+4+0-4+0-4+0-4

Possible rational root: ±1

Divide polynomial by (x-1): x^4+6x^3-3x^2-24x-4 = (x-1)(x^3+7x^2+4x+4x+4) = (x-1)(x^3+7x^2+8x+4)

Roots of the equation: x = 1, -2 ± i, where i = √(-1).

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