1.
Find a polynomial f(x) of degree 3 that has the following zeros.
4, 0, -7
Leave your answer in factored form.

The value of b as described in the task content by means of solving the two equations simultaneously is; b= -10.

The given simultaneous equations as indicated have been manipulated to determine the value of a which is; 0.5 in the task content.

Hence, by substituting a= 0.5 into the equation; 5 = 30a + b; we have;

5 = 30(0.5) + b

b = 5- 15 = -10.

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The required answers are calculated by using the simple interest formula:

a. She needs to deposit $793.65 in the account each month

b. The total money she put into an account for a year is $285,714.29

c. The total interest she earns is $514,285.71

The formula for the simple interest is

A = P(1 + RT)

Where,

A - amount after T years

P - principal amount

R - the rate of interest

T - time (years)

It is given that,

A = 8,00,000

T = 30 years

R = 6% = 0.06

So,

a. Finding the amount needs to deposit in the account each month:

We have A = P(1 + RT)

⇒ P = A/(1 + RT)

On substituting,

P = 8,00,000/(1 + 0.06×30)

  = 8,00,000/2.8

  = $285,714.29(per year)

Thus, the amount needs to deposit in the account for each month

= P/T×12

= 285,714.29/30×12

= $793.65

b. Finding the total money that she put into account:

That is nothing but,

P = A/(1 + RT)

On substituting,

P = 8,00,000/(1 + 0.06×30)

  = 8,00,000/2.8

  = $285,714.29(per year)

c. FInding the total interest:

We have I = A - P

⇒ I = 8,00,000 - 285,714.29

∴ I = $514,285.71

Therefore, a. $793.65, b. $285,714.29, and c. $514,285.71

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Answer:

The answer is G.

f-5 drop the class while s class is 3(f-5)

therefore, s=3(f-5)

The graphed system of inequalities is:

So the correct option is the third one.

On the graph we can see that the two lines are solid, so we need to use the symbols ≤ and ≥. (If instead, we had dashed lines, then we would need to use the symbols > and <).

We also can see that the region above the parabola is shaded (the shaded region represents the region of solutions for each inequality), and the region below the line is shaded, then the system of inequalities is of the form:

Only with that, we conclude that the correct option is the third option:

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From the constraint, we have

[tex]x+y+z=9 \implies z = 9-x-y[/tex]

so that [tex]s[/tex] depends only on [tex]x,y[/tex].

[tex]s = g(x,y) = xy + y(9-x-y) + x(9-x-y) = 9y - y^2 + 9x - x^2 - xy[/tex]

Find the critical points of [tex]g[/tex].

[tex]\dfrac{\partial g}{\partial x} = 9 - 2x - y = 0 \implies 2x + y = 9[/tex]

[tex]\dfrac{\partial g}{\partial y} = 9 - 2y - x = 0[/tex]

Using the given constraint again, we have the condition

[tex]x+y+z = 2x+y \implies x=z[/tex]

so that

[tex]x = 9 - x - y \implies y = 9 - 2x[/tex]

and [tex]s[/tex] depends only on [tex]x[/tex].

[tex]s = h(x) = 9(9-2x) - (9-2x)^2 + 9x - x^2 - x(9-2x) = 18x - 3x^2[/tex]

Find the critical points of [tex]h[/tex].

[tex]\dfrac{dh}{dx} = 18 - 6x = 0 \implies x=3[/tex]

It follows that [tex]y = 9-2\cdot3 = 3[/tex] and [tex]z=3[/tex], so the only critical point of [tex]s[/tex] is at (3, 3, 3).

Differentiate [tex]h[/tex] again and check the sign of the second derivative at the critical point.

[tex]\dfrac{d^2h}{dx^2} = -6 < 0[/tex]

for all [tex]x[/tex], which indicates a maximum.

We find that

[tex]\max\left\{xy+yz+xz \mid x+y+z=9\right\} = \boxed{27} \text{ at } (x,y,z) = (3,3,3)[/tex]

The second derivative at the critical point exists

[tex]$\frac{d^{2} h}{d x^{2}}=-6 < 0[/tex] for all x, which suggests a maximum.

Given, the constraint, we have

x + y + z = 9

⇒ z = 9 - x - y

Let s depend only on x, y.

s = g(x, y)

= xy + y(9 - x - y) + x(9 - x - y)

= 9y - y² + 9x - x² - xy

To estimate the critical points of g.

[tex]$&\frac{\partial g}{\partial x}[/tex] = 9 - 2x - y = 0

[tex]$&\frac{\partial g}{\partial y}[/tex] = 9 - 2y - x = 0

Utilizing the given constraint again,

x + y + z = 2x + y

⇒ x = z

x = 9 - x - y  

⇒ y = 9 - 2x, and s depends only on x.

s = h(x) = 9(9 - 2x) - (9 - 2x)² + 9x - x² - x(9 - 2x) = 18x - 3x²

To estimate the critical points of h.

[tex]$\frac{d h}{d x}=18-6 x=0[/tex]

⇒ x = 3

It pursues that y = 9 - 2 [tex]*[/tex] 3 = 3 and z = 3, so the only critical point of s exists at (3, 3, 3).

Differentiate h again and review the sign of the second derivative at the critical point.

[tex]$\frac{d^{2} h}{d x^{2}}=-6 < 0[/tex]

for all x, which suggests a maximum.

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Answer:

[tex]P(x) -0.1(x+1)(x-3)^2[/tex]

Step-by-step explanation:

So you can express a polynomial in factored form, where each factor of the polynomial is being multiplied by each other to get the original polynomial as such: [tex]p(x) = a(x+a)(x+b)(x+c)...[/tex] notice the a in front? Sometimes the value is 1, sometimes it's not, and I'm assuming it's not 1, since the y-intercept is given.

Anyways, when we express a polynomial in factored form: [tex]p(x) = a(x+a)(x+b)(x+c)...[/tex], it's pretty easy to see that -a, -b, and -c are roots, since if we plug in -a, the factor (x+a) becomes (-a + a) = 0, and since the factors are being multiplied by each other, the value p(-a) will be 0, since 0 * some value = 0.

The next thing to know is what multiplicity means. It essentially means when you have the same root "multiple times", or in other words it's the degree of the factor. For example: [tex]f(x) = (x+a)^3(x+b)^2(x+c) = (x+a)(x+a)(x+a)(x+b)(x+b)(x+c)[/tex], if you took each factor, you would see that the root -a shows up 3 times, thus the multiplicity is 3, which can also be determined by just looking at the initial degree... 3. The same thing goes for b and c, except for c, the degree isn't explicitly said, it's just 1.

Since we have a zero at x=3 one of the factors will be (x-3) since plugging in 3 to this makes it 0. But this has a multiplicity of 2, so the degree of the this factor is 2. So for we have the polynomial: [tex]p(x) = (x-3)^2[/tex]

Since we have a zero at x=-1, one of the factors will be at (x+1), since plugging in 1 into this makes the value 0. This has a multiplicity of 1, and you can write the degree 1, but if you don't explicitly write it, it can just be assumed to be 1, since a^1=a

So now we have the polynomial: [tex]p(x) = (x+1)(x-3)^2[/tex]

So let's just assume we have an "a" value in front as mentioned in the very beginning, we have the polynomial: [tex]p(x) = a(x+1)(x-3)^2[/tex] and we can solve for this value, using any point except for the zeroes. We can use the y-intercept given to solve for a

Plug in 0 as x (by definition of y-intercept) and set p(x) = -0.9

[tex]-0.9 = a(0+1)(0-3)^2[/tex]

Subtract/add values

[tex]-0.9 = a(1)(-3)^2[/tex]

Simplify

[tex]-0.9 = 9a[/tex]

Divide both sides by 9

-0.1 = a

So now let's plug this into our polynomial to get the final result!

[tex]P(x) -0.1(x+1)(x-3)^2[/tex]

Answer:

a) x + 10 = 45

b) 3 = 6 - x

c) d = s * t

Step-by-step explanation: This is actually pretty simple. Let's put down one theorem.

SYMMETRIC THEOREM OF PROPERTIES: If a equals b, then b must equal a. Thus, a = b and b = a.

This is all we're really doing.

Take a) for example, 45 = x + 10. Let's label (45) as A, and (x + 10) as B. Thus we have A = B. Now we know by the Symmetric Property, we have B = A. Thus, substituting B and A again, we get x + 10 = 45.

Just keep doing this with all the equations.

Hope this helped!

Answer: 4.6 (rounded to the nearest tenth)

Step-by-step explanation:

I assume that 1.6 is the coefficient of x, which is what we're trying to find out in this case.

Therefore, dividing 1.6 on both sides we get

from

1.6x = 7.37

to

x = 7.37/1.6 = 4.6 (approximately, rounded to the nearest tenth).

The probability that a person in this income bracket will be audited this year is of 3%.

A probability is given by the number of desired outcomes divided by the number of total outcomes.

In this problem, we have that 30 out of 1000 people are audited, hence the probability that a person in this income bracket will be audited this year is given by:

p = 30/1000 = 0.03 = 3%.

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The number of moles of hydrogen is 185 moles.

We know that the mole refers to the number of elementary entities that make up a substance. According to Avogadro, one mole of a substance contains 6.02 * 10^23 elementary entities which includes atoms, molecules and ions.

Now, we know the number of moles is obtained as the ratio of the mass to the molar mass of the substance.

Thus;

Mass of hydrogen = 370.g

Molar mass of hydrogen = 2 g/mol

Number of moles of hydrogen = 370.g/2 g/mol = 185 moles

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Answer:

$29765.50

Step-by-step explanation:

Median is the middlemost value (or the average of the 2 middlemost value if there are even number of values) in an ascending order.

First, arrange the salary list in an ascending order.

$21,971 , $25,997 , $27,512 , $32,019 , $38,860 , $43,702

As we can see here, there are even number of values (6), so the median will be the average of the 2 middlemost value, which will be:

Median = [tex]\frac{27512 + 32019}{2}[/tex]

= [tex]\frac{59531}{2}[/tex]

= $29765.50

Answer:

$29,765.50 is the median salary

Step-by-step explanation:

To find the median of the following salary list $32,019, $21,971, $27,512, $43,702, $38,860, $25,997

The steps to do this is:

Doing the steps:

This means take the average of $27,512 and $32,019

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Answer: 50 and 59

I divided 2950 by 25 because I knew it would somehow factor into that since it ends in 50. Then, I got 25 and 118. From there, I just divided 118 by 2 since I knew that it would end in a 9 and I multiplied 25 by 2. Not sure if this is really helpful, but you also try using a GCF calculator and look up all the factors of that number next time!

Answer:

Step-by-step explanation:

85!/82! = 85*84*83 = 592620

n choose 0 is always equals 1

n P n also always equals 1

Answer:

x = 5

Step-by-step explanation:

There's a Secant Theorem that someone else figured out (waaaay back in history) we just need to memorize it. So a secant is a line that touches the circle in two places. Your picture has two secants that both go thru the same point that's outside the circle. The secants each have a bit that's inside the circle and a bit that's outside the circle. And we could add together the inside and outside bits and get a total for the whole thing.

The secant theorem says that the outside piece × the whole thing on one secant = the outside piece × the whole thing on the other secant.

For the secant on top the "outside" bit is 9 and the whole thing is (2x+1+9). We'll times these together.

For the bottom secant the outside piece is 10 and the whole thing is (x+3+10). We'll multiply these together.

9(2x+1+9)=10(x+3+10)

simplify.

9(2x+10) = 10(x+13)

distribute.

18x + 90 = 101x + 130

combine like terms.

8x + 90 = 130

subtract 90

8x = 40

divide by 8

x = 5

see image.

The quartic equation behind the graph is y = - 2 · x⁴ + 4 · x³ + 6 · x² - 8 · x - 8.

Quartic functions are fourth grade polynomials and according to the fundamental theorem of algebra, such kind of expressions have at least two real roots and at most four real roots. The graph of the picture shows a polynomial with two roots of multiplicity 2: x₁ = - 1, x₂ = 2.

y = a · (x + 1)² · (x - 2)²      (1)

Where a is the leading coefficient.

If we know that (x, y) = (0, 4), then the leading coefficient of the polynomial is:

4 = a · (0 + 1) · (0 - 2)

4 = - 2 · a

a = - 2

Then, the quartic equation is equal to:

y = - 2 · (x + 1)² · (x - 2)²

y = - 2 · (x² + 2 · x + 1) · (x² - 4 · x + 4)

y = - 2 · [(x² + 2 · x + 1) · x² + (x² + 2 · x + 1) · (- 4 · x) + (x² + 2 · x + 1) · 4]

y = - 2 · (x⁴ + 2 · x³ + x² - 4 · x³ - 8 · x² - 4 · x + 4 · x² + 8 · x + 4)

y = - 2 · (x⁴ - 2 · x³ - 3 · x² + 4 · x + 4)

y = - 2 · x⁴ + 4 · x³ + 6 · x² - 8 · x - 8

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In order to make a 3-dimensional object into a 4-dimensional object, it's important to draw the 4-dimensional shape in a way that gives the illusion of the 3-dimensional object.

It should be noted that shapes play an important part in geometry.

Here, to make a 3-dimensional object into a 4-dimensional object, it's important to draw the 4-dimensional shape in a way that gives the illusion of the 3-dimensional object.

Also, it should be noted that a 4D tesseract can be used to project the image.

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Answer:

  A.  (-1, -4)

Step-by-step explanation:

The vertex can be found by converting the equation from standard form to vertex form.

Considering the x-terms, we have ...

  y = (x^2 +2x) -3

where the coefficient of x is 2. Adding (and subtracting) the square of half that, we get ...

  y = (x^2 +2x +(2/2)^2) -3 -(2/2)^2

  y = (x +1)^2 -4

Compare this to the vertex form equation ...

  y = a(x -h)^2 +k

which has vertex (h, k).

We see that h=-1 and k=-4. The vertex is (h, k) = (-1, -4).

On the attached graph, the vertex is the turning point, the minimum.

It should be noted that a trial balance is a report which lists the balances of the ledger accounts of a company.

It should be noted that the information is incomplete. Therefore, an overview will be given. It should be noted that the accounts that are reflected on a trial balance are related to the accounting items.

The adjusted trial balance simply lists the general ledger account balances after the adjustments have been made. In such a case, these adjustments typically include prepaid and accrued expenses, and non-cash expenses such as depreciation.

Furthermore, a trial balance is a list of the closing balances of ledger account while adjusted balance is a list of general account. An adjusted trial balance is typically prepared by creating a series of journal entries which are designed to account for any transactions that haven't been completed.

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Answer:

y= -3x+4

Step-by-step explanation:

y=mx+c

m= y2-y1 ÷ x2-x1

m= -3

y=-3x+c

7=-3(-1)+c

c=4

y=-3+4

Answer:

2.52 m

Step-by-step explanation:

let the height be h m.

let the shorter ladder be x m away from the base of wall.

(x+1.96)²+h²=5.6²

x²+h²=4.2²

subtract

(x+1.96)²-x²=5.6²-4.2²

(x+1.96+x)(x+1.96-x)=31.36-17.64

(2x+1.96)(1.96)=13.72

2x+1.96=13.72/1.96=7

2x=7-1.96=5.04

x=5.04/2=2.52

Using the law of sines, it is found that the distance between the kite and the dog is of 284.77 meters.

Suppose we have a triangle in which:

The lengths and the sine of the angles are related as follows:

[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]

For the situation described, we have that:

Hence:

[tex]\frac{\sin{42^\circ}}{h} = \frac{\sin{87^\circ}}{425}[/tex]

Applying cross multiplication:

[tex]h = 425\frac{\sin{42^\circ}}{\sin{87^\circ}}[/tex]

h = 284.77 meters.

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Answer:

135

Step-by-step explanation:

You can add the number which are in negative o

as well

The number is in descending order by 4.

Then add the first and last number i.e.

433+107===> 540

divide 540 by 4

540/4

= 135

a. Angle BAC is a central angle.

b. Arc BEC is a major arc.

c. Arc BC is a minor arc.

d. m(BEC) = 260°

e. m(BC) = 100°

An arc that has a measure that is greater than 180 degrees or is greater than a semicircle (half a circle) is referred to as a major arc.

An arc that has a measure that is less than 180 degrees or is not up to a semicircle (half a circle) is referred to as a minor arc.

A central angle can be described as an angle with a vertex at the center of a circle and has two radii of the circle at sides.

a. Angle BAC is a central angle.

b. Arc BEC is a major arc.

c. Arc BC is a minor arc.

d. m(BEC) = 360 - 100 [central angle theorem]

m(BEC) = 260°

e. m(BC) = angle BAC = 100° [central angle theorem]

m(BC) = 100°

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Solving the given logarithmic equation, it is found that the horn is 100 times more intense compared to the weakest sound heard to the human ear.

The equation is given by:

[tex]d = 10\log{\frac{p}{p_0}}[/tex]

In which:

In this problem, we have that d = 20, and we have to solve the logarithmic equation for p to find how many times more intense the sound is:

[tex]d = 10\log{\frac{p}{p_0}}[/tex]

[tex]20 = 10\log{\frac{p}{p_0}}[/tex]

[tex]\log{\frac{p}{p_0}} = 2[/tex]

The logarithm is inverse of the function [tex]10^x[/tex], hence we apply the function to both sides to find the ratio.

[tex]\frac{p}{p_0} = 10^2[/tex]

[tex]\frac{p}{p_0} = 100[/tex]

Hence, the horn is 100 times more intense compared to the weakest sound heard to the human ear.

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Answer:

a)

Step-by-step explanation:

Answer:

Step-by-step explanation:

C = 2 π r

A = π r²

~~~~~~~~~~~

(1). C = 8 π

Let the length of the arc ADB is "x"

x = ( 8π ÷ 360° ) × 340°

x = [tex]\frac{68}{9} \pi[/tex]

(2). A = 16 π

The area of shaded region is

( 16 π ÷ 360° ) × 340° = [tex]\frac{136}{9} \pi[/tex]

[tex]\bold{Answer} \downarrow[/tex]

[tex]Length\ of\ Arc\ ADB = \large\boxed{\frac{68\pi}{9} }[/tex]

[tex]Area\ of\ shaded\ region= \large\boxed{\frac{136\pi}{9} }[/tex]

Formulas needed:

[tex]Arc\ length=\boxed{2\pi r(\frac{\theta}{360} )}[/tex]

[tex]Area \ of\ a\ sector=\boxed{(\frac{n}{360})\times\pi \times r^2}[/tex]

[tex]Area\ of\ Circle=\boxed {\pi\times r^2}[/tex]

Step-by-step explanation:

First, let's find the arc length:

[tex]Arc\ length=2\pi r(\frac{\theta}{360})[/tex]

[tex]=2\pi (4)(\frac{340}{360})[/tex]

[tex]=2\pi (4)(\frac{17}{18})[/tex]

[tex]=2\pi(\frac{68}{18})[/tex]

[tex]=\frac{136\pi }{18}[/tex]

[tex]\longrightarrow \large\boxed{\frac{68\pi}{9} }[/tex]

Next, let's find the area of the sliver of the circle we just found the arc length of.

[tex]Area \ of\ a\ sector=(\frac{n}{360})\times\pi \times r^2[/tex]

[tex]=(\frac{20}{360} )\times \pi \times 4^2[/tex]

[tex]=(\frac{1}{18})\times \pi \times 16[/tex]

[tex]=\frac{16\pi}{18}[/tex]

[tex]\longrightarrow \frac{8 \pi}{9}[/tex]

Finally, find the area of the entire circle and subtract the sliver area by that.

[tex]Area\ of\ Circle= {\pi\times r^2[/tex]

[tex]=\pi \times 4^2[/tex]

[tex]\longrightarrow16\pi[/tex]

Subtracting the area by the sliver.

[tex]Area\ of\ entire\ circle \ -\ Area\ of\ sliver\ of\ circle[/tex]

[tex]=16\pi -\frac{8\pi}{9}[/tex]

[tex]=\frac{144\pi}{9} -\frac{8\pi }{9}[/tex]

[tex]\longrightarrow \large\boxed{\frac{136\pi}{9} }[/tex]

Check the picture below.

[tex]\textit{area of a trapezoid}\\\\ A=\cfrac{h(a+b)}{2}~~ \begin{cases} h~~=height\\ a,b=\stackrel{parallel~sides}{bases~\hfill }\\[-0.5em] \hrulefill\\ a=5\\ b=9\\ h=7 \end{cases}\implies \begin{array}{llll} A=\cfrac{7(5+9)}{2}\implies A=\cfrac{7(14)}{2} \\\\\\ A=49 \end{array}[/tex]

The equation the completes the proof in step 6 is m<w + m<x = m<x + m<z

In step 4 of the proof, we have the following equation

m<w + m<x = m<y + m<z

In step 5, we have

m<x = m<y

When m<x = m<y is substituted to the equation in step 4, we have:

m<w + m<x = m<x + m<z

Hence, the equation the completes the proof in step 6 is m<w + m<x = m<x + m<z

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