If
f (n)(0) = (n + 1)!
for
n = 0, 1, 2, ,
find the Maclaurin series for f.

The solution to the equation -4[x + 5] = -16 is x = -1.

To solve the equation -4[x + 5] = -16 for x, we will follow these steps:

First, we need to isolate the term with the variable. We can start by dividing both sides of the equation by -4 to get rid of the coefficient:

-4[x + 5] / -4 = -16 / -4

Simplifying both sides, we get:

x + 5 = 4

Next, solve for x by subtracting 5 from both sides:

x + 5 - 5 = 4 - 5

Simplifying this, we find:

x = -1

When [tex]x =  - 7, 2{x^2} + 10x - 20 = 8[/tex] and If [tex]2{x^2} + 10x -20[/tex] is divided by [tex]\left( {x + 7} \right)[/tex], the remainder is 12. Option (D) is correct and option (E) is correct.

Further Explanation:

Explanation:

If division of a polynomial by a binomial result in a remainder of zero means that the binomial is a factor of polynomial.

The synthetic division can be expressed as follows,

[tex]\begin{aligned}- 7\left| \!{\nderline {\,{2\,\,\,\,\,\,\,\,\,\,10\,\,\,\,\,\,\,\,\,\, - 20} \,}} \right.\hfill\\\,\,\,\,\,\,\underline {\,\,\,\,\,\,\,\,\,\, - 14\,\,\,\,\,\,\,\,\,\,\,\,\,\,28}  \hfill\\ \,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\, - 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,8 \hfill\\\end{aligned}[/tex]

The last entry of the synthetic division tells us about remainder and the last entry of the synthetic division is [tex]8[/tex]. Therefore, the remainder of the synthetic division is [tex]8[/tex].

When [tex]x =  - 7, 2{x^2} + 10x - 20 = 8[/tex] and If [tex]2{x^2} + 10x -20[/tex] is divided by [tex]\left( {x + 7} \right)[/tex], the remainder is 12. Option (D) is correct and option (E) is correct.

Option (A) is not correct.

Option (B) is not correct.

Option (C) is not correct.

Option (F) is not correct.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Synthetic Division

Keywords: division, factor (x+7), remainder 8, statements, true, apply, divided, binomial synthetic division, long division method, coefficients, quotients, remainders, numerator, denominator, polynomial, zeros, degree.

Answer:

true

Step-by-step explanation:

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Answer:

The system of equations is the option A

[tex]x+y=280[/tex]

[tex]y=4x[/tex]

Jenna bought [tex]56[/tex] parrot fish and [tex]224[/tex] trigger fish

Step-by-step explanation:

Let

x-----> the number of parrot fish

y-----> the number of trigger fish

we know that

[tex]x+y=280[/tex] -----> equation A

[tex]y=4x[/tex] ------> equation B

The answer Part 1) is the option A

Substitute equation B in equation A

[tex]x+4x=280[/tex]

Solve for x

[tex]5x=280[/tex]

[tex]x=56[/tex]

Find the value of y

[tex]y=4x[/tex]

[tex]y=4*56=224[/tex]

therefore

The answer part 2) is

Jenna bought [tex]56[/tex] parrot fish and [tex]224[/tex] trigger fish

Final answer:

The correct system of equations modeling Jenna's purchase of tropical fish, where x is the number of parrotfish and y is the number of triggerfish, is x + y = 280 and y = 4x, which is option A.

Explanation:

The problem given is about modeling a real-world situation with a system of equations where x represents the number of parrotfish Jenna bought and y represents the number of triggerfish she bought. Jenna bought 280 tropical fish for a museum display, and she bought 4 times as many triggerfish as parrotfish. The system of equations that models this situation is:

x + y = 280

y = 4x

This matches option A from the given choices. The first equation represents the total number of fish Jenna bought, while the second equation represents the relationship between the number of triggerfish and parrotfish, with the triggerfish being four times the number of parrotfish. To solve, substitute the second equation into the first to find the values of x and y.

Answer:

h(x) = 2x2 + 14x – 60      (D)

Step-by-step explanation:

The quadratic function that represents h(x) with roots at 3 and -10 is h(x) = x^2 - 7x - 30 since it correctly reflects the sum and product of its roots.

The question asks us to find which function could represent h(x), given that h(3) = h(–10) = 0. This means the quadratic function has roots at x = 3 and x = -10. A quadratic function with these roots can be written as h(x) = a(x - 3)(x + 10), where a is a non-zero coefficient.

By expanding this expression, we would get h(x) = a(x^2 + 7x - 30). We can then see if any of the given options matches this form upon factoring out the leading coefficient.

Comparing the given options:

Therefore, the correct function that could represent h(x) is h(x) = x^2 - 7x - 30.

Answer:

Length of the conjugate axis is 6 units.

Step-by-step explanation:

Given Equation is ,

[tex]\frac{(x-1)^2}{25}-\frac{(y+3)^2}{9}=1[/tex]

To find: Length of the conjugate axis.

We know that given equation is Equation of Hyperbola.

First Transverse Axis: Axis passing through the vertices is called the transverse axis. Length of the transverse axis is 2a.

Now, Conjugate Axis: Axis which is perpendicular to the transverse axis through the center is called the conjugate axis. Length of the Conjugate axis  is 2b.

Equation in standard form is written as ,

[tex]\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1[/tex]

So, Comparing with Standard equation,

we get, a² = 25 ⇒ a = 5   and     b² = 9 ⇒ b = 3

Thus, Length of the conjugate axis = 2 × 3 = 6 unit

Therefore, Length of the conjugate axis is 6 units.

For the given equation of hyperbola:

[tex]\dfrac{(x-1)^2}{25}-\dfrac{(y+3)^2}{9}=1[/tex]

The length of the conjugate axis is 6 units.

In analytic geometry, a hyperbola is a conic section created when a plane meets a double right circular cone at an angle that overlaps both cone halves.

The standard equation of the hyperbola is,

[tex]\dfrac{(x-h)^2}{a^2}- \dfrac{(y-k)^2}{b^2} =1[/tex]

Where,

a is half of the transverse axis.

b is the half of the conjugate axis.

The given equation is,

[tex]\dfrac{(x-1)^2}{25}-\dfrac{(y+3)^2}{9}=1[/tex]

Comparing it with the above standard equation, we get:

a² = 25

⇒ a = ± 5

b² = 9

⇒ b = ± 3

Since, the length of the conjugate axis of hyperbola = 2b

Therefore,

The length of the conjugate axis:

= 2x3

= 6 units.

Hence,

The required length of the conjugate axis is 6 units.

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Answer:

Option D - $6312.38

Step-by-step explanation:

Given : If you were to place $5,000 in a savings account that pays 6% interest compounded continuously.

To find : How much money will you have after 4 years?

Solution :

Using compound interest formula,

[tex]A=P(1+r)^t[/tex]

Where A is the amount,

P is the principle P=$5000  

r is the rate r=6%=0.06

t is the time t= 4 years

Substitute the value,

[tex]A=P(1+r)^t[/tex]

[tex]A=(5000)(1+0.06)^4[/tex]

[tex]A=(5000)(1.06)^4[/tex]

[tex]A=5000\times 1.262[/tex]

[tex]A=6312.38[/tex]

The money he have after 4 years is $6312.38

Therefore, Option D is correct.

Answer:

A. $6,356.25

Step-by-step explanation:

To calculate the amount of money you will have after 4 years in a savings account with continuous compounding, we can use the formula:

A = P * e^(rt)

Where:

A = the amount of money accumulated after t years

P = the initial principal amount (in this case, $5,000)

e = the base of the natural logarithm (approximately 2.71828)

r = the annual interest rate (in decimal form, so 6% becomes 0.06)

t = the number of years (in this case, 4)

Plugging in the values, we get:

A = 5000 * e^(0.06 * 4)

Using a calculator, we can evaluate this expression:

A ≈ 5000 * 2.71828^(0.24)

A ≈ 5000 * 1.276281

A ≈ 6381.405

Rounding to the nearest cent, the amount of money you will have after 4 years is approximately $6,381.41.

Therefore, the closest answer option is A. $6,356.25.

Answer: False

Step-by-step explanation:

n/7 - 8 < -11

add 8 to each side

n/7 < -3

multiply by 7

n< -21

False

The only time we flip the inequality symbol is when we multiply or divide by a negative.

Hope this helps!

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To find the point P on the y-axis that is on both planes, set x = 0 and z = 0 for each plane and solve for y. The unique point P is (0, 1, 0). To find a unit vector u parallel to both planes, find the normal vectors of the planes and normalize one of them. The unit vector u with a positive first coordinate is (2/sqrt(30), 1/sqrt(30), 5/sqrt(30)). The vector equation for the line of intersection of the two planes is r(t) = (0, 1, 0) + t(5, -10, -2).

To find the unique point P on the y-axis that is on both planes, we need to find the values of x, y, and z that satisfy the equations of both planes.

For the first plane, 2x + y + 5z = 1, if we set x = 0 and z = 0, we can solve for y:

0 + y + 0 = 1
y = 1

Therefore, the point P on the y-axis that is on the first plane is (0, 1, 0).

For the second plane, 2x + 5z = 0, since there is no y term, any point on the y-axis will satisfy this equation. So, we can choose any value for y and set x = 0 and z = 0. Let's choose y = 1:

2(0) + 5(0) = 0

Therefore, the point P on the y-axis that is on the second plane is (0, 1, 0).

Since both points have the same x and z coordinates, the unique point P on the y-axis that is on both planes is (0, 1, 0).

To find a unit vector u with a positive first coordinate that is parallel to both planes, we can find the normal vectors of the planes and then normalize one of them.

For the first plane, the normal vector is (2, 1, 5). For the second plane, the normal vector is (2, 0, 5). Both normal vectors are parallel to the planes.

Let's normalize the first vector:

||u|| = sqrt(2^2 + 1^2 + 5^2) = sqrt(30)

So, the unit vector u with a positive first coordinate that is parallel to both planes is (2/sqrt(30), 1/sqrt(30), 5/sqrt(30)).

To find a vector equation for the line of intersection of the two planes, we can take the cross product of the normal vectors of the planes to get a vector that is perpendicular to both planes.

Let's calculate the cross product:

(2, 1, 5) x (2, 0, 5) = (5, -10, -2)

Now, we can use one of the points on the line of intersection, which is (0, 1, 0), and the direction vector (5, -10, -2) to form the vector equation:

r(t) = (0, 1, 0) + t(5, -10, -2)

7x10^-5 as a decimal is 0.00007, achieved by multiplying 7 by 0.00001, which represents 10^-5.

To convert 7x10^-5 into decimal form, you can use the concept that a negative exponent indicates the inverse. This means that 10^-5 is the same as 1/10^5 or 1 divided by 100,000. Thus, to find 7x10^-5 in decimal form, you multiply 7 by 0.00001.

Here's the calculation:

7 x 0.00001 = 0.00007

Therefore, 7x10^-5 in decimal form is 0.00007.

Answer:

It’s true

Step-by-step explanation: just took the test

Using the slope-intercept form, the slope and y-intercept of the line y = x – 5 is D) slope: 1, y-intercept: –5.

The slope is the ratio of the vertical changes to the horizontal changes between two points of the line.

The slope of the line is calculated as follows:

m = Δy/Δx

Given that the line y = x – 5

Let us consider

y = x – 5

Therefore, the slope of the line is;

m = 1

y-intercept of the line.

b = -5

Hence, the slope and y-intercept of the line y = x – 5 is D) slope: 1, y-intercept: –5

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The reduced price of the object is equal to $12.

The percentage is defined as representing any number with respect to 100. It is denoted by the sign %. The percentage stands for "out of 100." Imagine any measurement or object being divided into 100 equal bits.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

Given that the price of an item has been reduced by 85%. The original price was $80.

The reduced price of the item will be calculated as,

The reduction in price would be:

80 x 0.85 = 68

Price after discount is calculated as,

80 - 68 = $12

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One of the factors of 54xy + 45x + 18y + 15 is b) 6y + 5.

The GCF of two or more than two numbers is the highest number that divides the given two numbers completely.

We also know that distributive property states a(b + c) = ab + ac.

Given,

54xy + 45x + 18y + 15.

= 54xy + 18y + 45x + 15.

= 18y(3x + 1) + 15(3x + 1).

= (3x + 1)(18y + 15).

= (3x + 1)(3)(6y + 5).

So, a factor of 54xy + 45x + 18y + 15 is (6y + 5).

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Answer:  [tex]\dfrac{8}{9}[/tex]

Step-by-step explanation:

Given : Marty has saved $72. He spent $8 on a video rental.

i.e. The amount of money he saved = $72

The amount of money he spent on video rental = $8

Now, the amount of money he had left = [tex]\$72-\$8=\$64[/tex]

Thus , the ratio that represent what portion of his savings he has left :-

[tex]\dfrac{64}{72}=\dfrac{8}{9}[/tex]

Hence, the ratio that represent what portion of his savings he has left :-

[tex]=\dfrac{8}{9}[/tex]