Write the polynomial in standard form with roots of 1/4 and +5i

To get 51 using the numbers 8, 5, 6, and 7 exactly once each, you can use the following mathematical expression:

(8 x 6) - 7 + 5 = 51

How it works:

1. Multiply 8 by 6 to get 48: (8 x 6) = 48

2. Subtract 7 from 48 to get 41: 48 - 7 = 41

3. Add 5 to 41 to get 51: 41 + 5 = 51

Therefore, (8 x 6) - 7 + 5 = 51.

Answer:

Step-by-step explanation:

8 x (7 - 5) + 6 = 51

1. The statement "Each exterior angle of a regular hexagon is acute" is True.

2. The statement "The sum of the interior angles of a polygon is always a multiple of 180" is False.

3. The statement "In any polygon, the larger the number of vertices, the smaller the measure of an exterior angle" is True.

1. TRUE: Each exterior angle of a regular hexagon is acute.
A regular hexagon has six equal sides and six equal interior angles. The sum of the interior angles of a hexagon is (6-2) * 180 = 720 degrees. Since it's a regular hexagon, each interior angle is 720/6 = 120 degrees. The exterior angles are supplementary to the interior angles, so each exterior angle is 180 - 120 = 60 degrees. Since 60 degrees is less than 90 degrees, each exterior angle is acute.

2. FALSE: The sum of the interior angles of a polygon is always a multiple of 180.
The formula for the sum of the interior angles of a polygon is (n-2) * 180, where n is the number of vertices (or sides). As you can see, the result is always a multiple of 180.

3. TRUE: In any polygon, the larger the number of vertices, the smaller the measure of an exterior angle.
For a regular polygon, the measure of an exterior angle can be calculated as 360/n, where n is the number of vertices (or sides). As the number of vertices increases, the measure of an exterior angle decreases, since they are inversely proportional.

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The measure of angle JML is 180 degrees because in a circle, an angle formed by two chords intersecting inside the circle.

In a circle, the measurement of angle formed by two chords intersecting inside the circle is half the sum of the arcs intercepted by the angle. Using this property, we can find the measure of angle JML.

Since angle JKL is a right angle, its intercepted arc is the diameter of the circle. Therefore, its measure is 180 degrees.

By the same property, we know that angle JML is half the sum of the arcs intercepted by it. The arcs intercepted by angle JML are arcs JL and KM.

Since angle JKL is a right angle, arc JL is also 180 degrees.

Since J, K, L, and M are concyclic, we know that angle JKM is supplementary to angle JLM. Therefore, arc KM is the supplement of arc JL and has measure 360 - 180 = 180 degrees.

Thus, the sum of the intercepted arcs is 180 + 180 = 360 degrees, and angle JML is half of this, so its measure is 180 degrees.

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It appears to be setting up a proportion between the central angle of the sector and the ratio of arc length to the circumference of the circle, rather than the ratio of the central angle to the full angle of the circle. Then the area should equal  to 126.9 [tex]ft^2.[/tex]

To find the area of sector XZY, we need to know the measure of angle XYZ. However, the given equation n/360 = 115/225 is incorrect, as it appears to be setting up a proportion between the central angle of the sector and the ratio of arc length to the circumference of the circle, rather than the ratio of the central angle to the full angle of the circle.

To find the correct measure of angle XYZ, we need to use the formula:

Area of sector XZY = (n/360) x π[tex]r^2[/tex]

where r is the radius of circle Z.

We know that the area of circle Z is 255 square feet, so we can find the radius as follows:

Area of circle Z = π[tex]r^2[/tex]

255 = π[tex]r^2[/tex]

[tex]r^2[/tex] = 81

r = 9

Now we can solve for n using the given ratio of 115/225:

n/360 = 115/225

n = (115/225) x 360

n = 184.32

Rounding to the nearest tenth, we get:

n ≈ 184.3

Finally, we can find the area of sector XZY as:

Area of sector XZY = (n/360) x π[tex]r^2[/tex]

Area of sector XZY = (184.3/360) x π[tex](9)^2[/tex]

Area of sector XZY ≈ 126.9 [tex]ft^2[/tex]

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Answer:

The answer is -8

Step-by-step explanation:

Here's a key

()= Do first

/ = Divide

If something is outside the () and there is no symbol then multiply.

The simple interest earned on $1,272 deposited in a bank for 20 years at 3% annual interest rate is $764.40.

To calculate the simple interest earned on a principal amount deposited in a bank, we use the formula:

Simple Interest = (Principal) x (Rate) x (Time)

where the rate is the annual interest rate, and the time is the number of years the money is deposited.

In this case, the principal is $1,272, the rate is 3%, and the time is 20 years.

Plugging in these values into the formula, we get:

Simple Interest = (1272) x (0.03) x (20)

Simple Interest = $764.40

Therefore, the simple interest earned on $1,272 deposited in a bank for 20 years at 3% annual interest rate is $764.40.

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The solutions to the system of nonlinear equations are

x = 9 and x =[tex]3^2[/tex]= 9.

To determine the solution to the system of nonlinear equations:

f(x) = g(x)

We can substitute the given expressions for f(x) and g(x) and simplify:

log3(x) + 3 =[tex]log3(x^3) - 1[/tex]

Using the properties of logarithms, we can simplify this equation as follows:

log3(x) + 3 = 3*log3(x) - 1

4 = 2*log3(x)

2 = log3(x)

x =[tex]3^2[/tex]

Therefore, the solutions to the system of nonlinear equations are x = 9 and [tex]x = 3^2 = 9.[/tex]

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Triangle PQR is not a right triangle.

To determine whether triangle PQR is a right triangle, we need to check if any of its angles is a right angle (90 degrees). We can use the slope formula to find the slopes of the sides of the triangle and check if any of the slopes are negative reciprocals (perpendicular) to each other.

Let's calculate the slopes of the sides PQ, QR, and RP:

Slope of PQ = (y₂ - y₁) / (x₂ - x₁)

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

= 1 / 3

Slope of QR = (y₂ - y₁) / (x₂ - x₁)

= (-4 - 2) / (5 - 3)

= -6 / 2

= -3

Slope of RP = (y₂ - y₁) / (x₂ - x₁)

= (1 - (-4)) / (0 - 5)

= 5 / (-5)

= -1

Now, let's check if any of the slopes are negative reciprocals of each other. We can compare the products of the slopes:

Product of PQ slope and QR slope = (1/3) * (-3) = -1

Product of QR slope and RP slope = (-3) * (-1) = 3

Product of RP slope and PQ slope = (-1) * (1/3) = -1/3

Since the product of the slopes of QR and RP is not equal to -1, triangle PQR is not a right triangle.

Therefore, triangle PQR is not a right triangle.

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Answer:

A regular n-gon with one interior angle of 150° has 12 sides.

Step-by-step explanation:

The formula for the measure of each interior angle of a regular n-gon is:

where:

We are given that one interior angle measures 150°, so we can set up the equation:

Multiplying both sides by n, we get:

Distributing, we get:

Subtracting 150n from both sides, we get:

Adding 360 to both sides, we get:

Dividing both sides by 30, we get:

Therefore, a regular n-gon with one interior angle of 150° has 12 sides.

The ratio table shows the costs for different amounts of birdseed. So, the unit rate is $2 per pound in this example.

To find the unit rate in dollars per pound, follow these steps:
1. Look at the given ratio table, which shows the costs for different amounts of bird seed.
2. Identify the cost and the corresponding amount (in pounds) of bird seed for any one row in the table.
3. Divide the cost (in dollars) by the amount (in pounds) to calculate the unit rate.

For example, if the table shows that the cost is $6 for 3 pounds of bird seed, the unit rate would be calculated as follows:

Unit rate = Cost / Amount
Unit rate = $6 / 3 pounds
Unit rate = $2 per pound
So, the unit rate is $2 per pound in this example

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We get that the plane is about 2193.0 feet from the airport, Rounding to the nearest tenth

To solve the problem, we will use trigonometry and the tangent function, which relates the other facet of a right triangle to the adjoining aspect:

tan(theta) = opposite / adjacent

wherein theta is the angle of descent, opposite is the change in height, and adjacent is the space from the airplane to the airport.

Rearranging the formula, we get:

adjacent = contrary / tan(theta)

because the angle of descent is 13 ranges and the alternate in height is from 500 ft, we've got:

contrary = 500 ft

theta = 13 stages

Substituting these values into the formula, we get:

adjacent = 500 ft / tan(13 ranges)

using a calculator, we find that tan(13 stages) is about 0.228, so:

adjacent = 500 feet / 0.228 = 2192.98 feet

Therefore, we get that the plane is about 2193.0 feet from the airport.

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The constant of proportionality between y and x in the graph is 3

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

The graph

On the graph, we have the following readings

(x, y) = (1, 3)

Using the above as a guide, we have the following:

The constant of proportionality between y and x in the graph is

k = y/x

substitute the known values in the above equation, so, we have the following representation

k = 3/1

Evaluate

k = 3

Hence, the constant of proportionality between y and x in the graph is 3

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The current scores of both Jenna and Kera in the game they were playing are

Jenny's current score is -15.

Kera's current score is 8.

In the given problem, we are given two players, Jenny and Kera, playing the game.

Jenny has -10 points, which means she already has negative points. He has since lost 5 more points. So his current score would be:

-10 - 5 = -15

So now Jenny has a score of -15.

Kera, meanwhile, has 22 points and 14 points to lose. So his current score would be:

22 - 14 = 8

So now Kera has 8 points.

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The instantaneous rate of change of cost with respect to x when x = 100 is 4.

We can begin by calculating C(100+h) and C(100):

C(100+h) = 0.000002(100+h)^3 + 4(100+h) + 300

C(100+h) = 0.000002(1,000,000 + 300h^2 + 30h^2 + h^3) + 400 + 4h + 300

C(100+h) = 0.000002h^3 + 0.0006h^2 + 4h + 700

C(100) = 0.000002(100)^3 + 4(100) + 300

C(100) = 2 + 400 + 300

C(100) = 702

Therefore,

C(100+h) - C(100) = (0.000002h^3 + 0.0006h^2 + 4h + 700) - 702

C(100+h) - C(100) = 0.000002h^3 + 0.0006h^2 + 4h - 2

Now, we can find the rate of change of cost with respect to x by dividing this expression by h and taking the limit as h approaches 0:

(C(100+h) - C(100))/h = (0.000002h^3 + 0.0006h^2 + 4h - 2)/h

(C(100+h) - C(100))/h = 0.000002h^2 + 0.0006h + 4 - (2/h)

As h approaches 0, the term 2/h approaches infinity, which means the rate of change of cost with respect to x is undefined. However, we can calculate the limit of the expression as h approaches 0 from the left and from the right to see if it has a finite value:

limit (h->0+) ((C(100+h) - C(100))/h) = 4

limit (h->0-) ((C(100+h) - C(100))/h) = 4

Since the left and right limits are equal, the overall limit exists and equals 4. Therefore, the instantaneous rate of change of cost with respect to x when x = 100 is 4.

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The calculated two double inequalities that define shaded region are 1 ≤ y < 5 and -3 < x ≤ 2

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

The graph

On the graph, we have the following properties

Using the above as a guide, we have the following:

1 ≤ y < 5

-3 < x ≤ 2

Hence, the two double inequalities that define shaded region are 1 ≤ y < 5 and -3 < x ≤ 2

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A plane flying just above the equator around the world 3 1/2 times would cover a distance of 4.375 x 10⁴ miles.

If an airplane travels around the world just above the equator once, it covers a distance of 1.25 x 10⁴ miles. To find how many miles it would cover if it flew around the world 3 1/2 times, we need to multiply this distance by 3.5:

1.25 x 10⁴ miles x 3.5 = 4.375 x 10⁴ miles

To understand this calculation, we need to know that 3 1/2 times means 3.5 times. So, we multiply the distance covered in one round of the world by 3.5 to find the total distance covered in 3 1/2 times around the world.

We use standard form to express the answer in a more compact and convenient way, where 4.375 x 10⁴ represents the number 43,750 in scientific notation.

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Answer:

Step-by-step explanation:

Let's call the total volume of the container "V".

We know that the container was originally 15% full, so the amount of water in the container was 0.15V.

When 48 litres of water was added, the new volume of water in the container became 0.15V + 48.

We also know that the container is now 75% full, so the new volume of water in the container must be 0.75V.

We can set up an equation to solve for V:

0.15V + 48 = 0.75V

Subtracting 0.15V from both sides:

48 = 0.6V

Dividing both sides by 0.6:

V = 80

So the container can hold 80 litres of water when it is full.

Answer:

x=2

Step-by-step explanation:

2x+y=10

3x=y

2x+3x=10

5x=10

x=2

If the triangle is isosceles, then the measure of the third angle could be 42 degrees

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

The sum of angles in a triangle is 180 degrees

If the triangle is isosceles, then we have the following possible sum of angles

Sum 1 = 42 + 96 + 96 = 234 -- false

Sum 1 = 42 + 96 + 42 = 180 -- true

Hence, if the triangle is isosceles, then the measure of the third angle could be 42 degrees

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If Holly cuts 6 ribbons into fifths for her craft project, after cutting, she has a total of 30 one-fifth-size ribbons (6 ribbons x 5 cuts each).

Holly has cut 6 ribbons into fifths for her craft project. This means that she has a total of 30 ribbons, each with a size of one-fifth. To understand this better, we can break it down into fractions. Each ribbon is one-fifth of a whole ribbon, and since Holly has cut 6 ribbons, she has 6 times one-fifth, which equals 30. So, to answer the question, Holly has 30 ribbons, each with a size of one-fifth. These ribbons can be used for various crafts, such as creating bows, wrapping presents, or decorating cards. The possibilities are endless, and with 30 ribbons, Holly can get creative and make a lot of beautiful crafts.

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The variance of the data set is 0.318652.

To find the variance of this data set, we need to use the formula:
variance = Σ(x - μ)² / n
Where Σ represents the sum, x is the value of each data point, μ is the mean value, and n is the number of data points.

Using this formula, we can calculate the variance as follows:

Variance = [(5.07 - 4.312)² + (3.57 - 4.312)² + (5.32 - 4.312)² + (3.19 - 4.312)² + (3.49 - 4.312)² + (4.25 - 4.312)² + (4.76 - 4.312)² + (5.19 - 4.312)² + (3.94 - 4.312)² + (4.34 - 4.312)²] / 10

Variance = [0.225769 + 0.449769 + 0.370656 + 1.323856 + 0.716164 + 0.006544 + 0.175684 + 0.382884 + 0.135556 + 0.000484] / 10

Variance = 3.18652 / 10

Variance = 0.318652

Therefore, the variance of the data set is 0.318652.

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The measure of angle PQR is 20 degrees.

We have,

The measure of angle PQR can be found by using the property that the measure of an inscribed angle is half the measure of its intercepted arc.

Given that the measure of arc QR is 40 degrees, we can conclude that the measure of angle PQR is half of that, which is:

The measure of angle PQR = 1/2 x Measure of arc QR

= 1/2 x 40 degrees

= 20 degrees

Therefore,

The measure of angle PQR is 20 degrees.

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The expression for the area of the rectangle inscribed in a circle of radius 9, in terms of n, is:

Area = n * √(81 - n^2)

To find an expression for the area of a rectangle inscribed in a circle of radius 9, we need to use the terms "rectangle" and "circle."

First, let's denote the length of the rectangle as "n" and the width as "w." Since the rectangle is inscribed in a circle with a radius of 9, its diagonal is equal to the diameter of the circle, which is 2 * 9 = 18.

Using the Pythagorean theorem for the right triangle formed by half of the diagonal, the length, and the width, we can write the equation:

(1/2 * 18)^2 = n^2 + w^2
81 = n^2 + w^2

Now, we need to express w in terms of n. To do this, we'll isolate w from the equation:

w^2 = 81 - n^2
w = √(81 - n^2)

The area of the rectangle can be calculated as the product of its length and width:

Area = n * w
Area = n * √(81 - n^2)

So, the expression for the area of the rectangle inscribed in a circle of radius 9, in terms of n, is:

Area = n * √(81 - n^2)

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The area of the whole figure is 31.5 sq. units.

The area of a given figure connotes its expanse in a 2 dimensional plane. The shape and size of a given figure determines how to calculate its area.

From the given question, the figure given can be likened to a rhombus. So that;

area of a rhombus = (diagonal 1 * diagonal 2)/ 2

Then,

area of the figure = (diagonal 1 * diagonal 2)/ 2

where: diagonal 1 = 7.5, and diagonal 2 = 8.4

So that;

area of the figure = (7.5*8.4)/ 2

                             = 63/ 2

                             = 31.5

The area of the whole figure is 31.5 sq. units.

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To find the interval of convergence of the power series, we can use the ratio test:

lim (n->inf) |((n+1)!(x+5)^(n+1)) / (1.3.5....(2n+1))| / |(n!(x+5)^n) / (1.3.5....(2n-1))|

= lim (n->inf) |(x+5)(2n+1)| / (2n+2) = |x+5| lim (n->inf) (2n+1)/(2n+2) = |x+5|

So the series converges if |x+5| < 1, and diverges if |x+5| > 1. Thus the interval of convergence is (-6, -4).

To check for convergence at the endpoints, we can use the limit comparison test with the divergent series:

1/1.3 + 1/1.3.5 + 1/1.3.5.7 + ... = sum (2n-1) terms = inf

At x = -6, we have:

sum (n=0 to inf) (n!(-1)^n)/(1.3.5....(2n-1)) = 1/1 - 1/1.3 + 1/1.3.5 - 1/1.3.5.7 + ... = inf

Since the series diverges at x = -6, the interval of convergence is (-6, -4] using set notation.

At x = -4, we have:

sum (n=0 to inf) (n!(1)^n)/(1.3.5....(2n-1)) = 1/1 - 1/1.3 + 1/1.3.5 - 1/1.3.5.7 + ... = 1 - 1/3 + 1/15 - 1/105 + ...

This is an alternating series that satisfies the conditions of the alternating series test, so it converges. Thus the interval of convergence is (-6, -4] using set notation, or [-6, -4) using interval notation.
To find the interval of convergence of the power series, we'll use the Ratio Test, which states that if the limit L = lim(n→∞) |aₙ₊₁/aₙ| < 1, then the series converges. Here, the series is given by:

Σ(n!(x+5)^n) / 1 . 3 . 5 ... (2n-1)

Let's find the limit L:

L = lim(n→∞) |(aₙ₊₁/aₙ)|
 = lim(n→∞) |((n+1)!(x+5)^(n+1))/(1 . 3 . 5 ... (2(n+1)-1)) * (1 . 3 . 5 ... (2n-1))/(n!(x+5)^n)|

Now, simplify the expression:

L = lim(n→∞) |(n+1)(x+5)/((2n+1))|

For the series to converge, we need L < 1:

|(n+1)(x+5)/((2n+1))| < 1

As n approaches infinity, the above inequality reduces to:

|x+5| < 1

Now, to find the interval of convergence, we need to solve for x:

-1 < x + 5 < 1
-6 < x < -4

The interval of convergence is given by the interval notation (-6, -4). To check the endpoints, we need to substitute x = -6 and x = -4 back into the original series and use other convergence tests such as the Alternating Series Test or the Integral Test. However, the power series will diverge at the endpoints, as the terms do not approach 0. Therefore, the interval of convergence remains (-6, -4).

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Each plant should produce 576.92 units and 384.61 units respectively to maximize the company's profit.

To maximize the company's profit, we need to find the quantity that maximizes the difference between the total revenue and the total cost. The total revenue is given by:

TR = pq

= (40 - 0.04q)(q1 + q2)

= 40q1 + 40q2 - 0.04[tex]q1^2[/tex]- 0.04[tex]q2^2[/tex] - 0.04q1q2

The total cost is given by:

TC = C1 + C2

[tex]= 6.7 + 0.03q1^2 + 7.9 + 0.04q2^2= 14.6 + 0.03q1^2 + 0.04q2^2[/tex]

The profit is given by:

π = TR - TC

= [tex]40q1 + 40q2 - 0.04q1^2 - 0.04q2^2 - 0.04q1q2 - 14.6 - 0.03q1^2 - 0.04q2^2[/tex]

Simplifying, we get:

π = [tex]40q1 + 40q2 - 0.04q1^2 - 0.04q2^2 - 0.04q1q2 - 14.6 - 0.03q1^2 - 0.04q2^2[/tex]

= [tex]-0.03q1^2 - 0.04q2^2 - 0.04q1q2 + 40q1 + 40q2 - 14.6[/tex]

To maximize profit, we need to take the partial derivatives of the profit function with respect to q1 and q2 and set them equal to zero:

∂π/∂q1 = -0.06q1 - 0.04q2 + 40 = 0

∂π/∂q2 = -0.08q2 - 0.04q1 + 40 = 0

Solving these equations simultaneously, we get:

q1 = 576.92

q2 = 384.61

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Step-by-step explanation:

you can solve cos(u) by

cos(u) = adjecent / hypotenes...general formula of cos

cos(u) = √44 / 12

cos(u) = 2√11 / 12 ..... √44 = √4×11 = 2√11

cos(u) = √11 / 6

u = cos^-1 ( √11 / 6 ) ..... divided both aide by cos ( multiple by cos invers )

u = 56.442 .... so we get it's angle

Answer:

[tex]cos(U)=\frac{\sqrt{11} }{6}[/tex]

Step-by-step explanation:

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. Therefore, we have:

cos(U) = adjacent/hypotenuse = TU/SU

We are given that TU = sqrt(44) and SU = 12, so:

cos(U) = sqrt(44)/12

To simplify this expression, we can first factor 44 into 4 * 11, since 4 is a perfect square and a factor of 44:

cos(U) = sqrt(4 * 11) / 12

cos(U) = (sqrt (4) * sqrt (11)) / 12

cos (U) = (2 * sqrt (11)) / 12

Simplifying the fraction by dividing both the numerator and denominator by 2, we get:

cos(U) = sqrt(11)/6

Therefore, the exact value of cos(U) in simplest radical form is sqrt(11)/6

Furthermore, if you want another way to write the answer, dividing by 6 is the same as multiplying by 1/6 so you can do cos (U) = 1/6 * sqrt (11)

Although the other individual was correct that you use inverse trig (cos ^ -1) to find the measure of U, getting an exact answer requires us to leave it in simplest radical form since the number is so large and at best will yield an approximation if you don't keep it in simplest radical form.

i hope this helps you.

The solution of the composite function, (f + g)(x)  is 8x + 7

A function relates input and output. In other words, a function is a special relationship among the inputs (independent variable) and their outputs (dependent variable).

A composite function is a function that depends on another function.

Therefore, let's solve the composite function

f(x) = x + 3

g(x) = 7x + 4

Hence,

(f + g)(x) can be solved as follows:

(f + g)(x)  = f(x) + g(x)

(f + g)(x)  = x + 3 + 7x + 4

combine like terms

(f + g)(x)  = x + 7x + 3 + 4

(f + g)(x)  = 8x + 7

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