The polynomial function p(x)=x3−4x2+ax+20 has (x+1) as one of its factors.

What would be the remainder when p(x) is divided by (x+2)

By using the De Moivre's formula, the quartic roots of the complex numbers (in polar form) are z₁ = (√2, π/6), z₂ = (√2, 2π/3), z₃ = (√2, 7π/6), z₄ = (√2, 5π/3).

Complex numbers are numbers of the form a + i b, where a and b are the real and imaginary component, respectively. In other words, complex numbers are an expansion from real numbers. The n-th root of a complex number is found by using the De Moivre's formula:

[tex]\sqrt[n]{z} = \sqrt[n]{r}\cdot \left[\cos \left(\frac{\theta + 2\pi\cdot k}{n} \right) + i\,\sin \left(\frac{\theta + 2\pi\cdot k}{n}\right)\right][/tex], for i = {0, 1, 2, ...,  n - 1}.

Where:

The norm of the complex number is found by Pythagorean theorem:

[tex]r = \sqrt{(2\sqrt{3})^{2}+(-2)^{2}}[/tex]

r = 4

And the direction is determined below:

[tex]\theta = \tan^{-1} \left(\frac{2\sqrt{3}}{-2} \right)[/tex]

θ = 2π/3 rad

Then, the quartic roots of the complex numbers (in polar form) are z₁ = (√2, π/6), z₂ = (√2, 2π/3), z₃ = (√2, 7π/6), z₄ = (√2, 5π/3).

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Using an exponential function, it is found that the value of David's Investment will be at least $415 after a period of 20 years.

An increasing exponential function is modeled by:

[tex]A(t) = A(0)(1 + r)^t[/tex]

In which:

In this problem, the initial value and the growth rate for his investment are given by:

A(0) = 230, r = 0.03.

Hence, the value after t years is given by:

[tex]A(t) = 230(1.03)^t[/tex]

Then, the value will be of at least $415 when:

[tex]A(t) \geq 415[/tex]

[tex]230(1.03)^t \geq 415[/tex]

[tex](1.03)^t \geq \frac{415}{230}[/tex]

[tex]\log{(1.03)^t} \geq \log{\frac{415}{230}}[/tex]

[tex]t\log{1.03} \geq \log{\frac{415}{230}}[/tex]

[tex]t \geq \frac{\log{\frac{415}{230}}}{\log{1.03}}[/tex]

[tex]t \geq 20[/tex]

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Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

The angle sum theorem tells you the sum of angles of a triangle is 180°. Angles of a linear pair are supplementary, so total 180°. These facts can be used to write a relation that can be solved for x.

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The interior angle at the right end of the line is the supplement of 4x:

  180° -4x

The interior angle at the left end of the line is the supplement of 2x:

  180° -2x

  86° +(180° -4x) +(180° -2x) = 180°

Then the solution to the triangle is ...

  266° = 6x . . . . . . . . . add 6x-180°

  x = 266°/6 = 44 1/3° . . . . . divide by 6

The equation represents a circle whose center is (-4, 1) and has a radius r = √2

We have the equation:

[tex]x^2 + y^2 + 8x - 2y + 15 = 0[/tex]

We can rewrite this as:

[tex](x^2 + 8x ) + (y^2 - 2y ) + 15 = 0\\\\\\\\(x^2 + 8x + 16) - 16 + (y^2 - 2y + 1) - 1 + 15 = 0\\\\(x + 4)^2 + (y - 1)^2 - 16 - 1 + 15 = 0\\\\(x + 4)^2 + (y - 1)^2 = 2[/tex]

Where we just completed squares.

Comparing this with the general circle equation, we can see that the center is the point (-4, 1) and the radius is √2.

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Answer:

Should be 60°

Step-by-step explanation:

Since each interior angle in an octagon is 120°, you would subtract 120° from 180° (which is the angle of a line). This would get you 60 °.

Answer: they can conclude the distributions are different at the 95% confidence interval. Compute the z test statistic to solve this.

Step-by-step explanation: to get the 95% confidence interval on the mean take the mean value (984) +/- (2*standard deviation divided by the square root of the number of samples) (I.e. 984 +/- (2*267/sqrt(501)))

Answer:

[tex]f(x)=x^3 +4x^2 +16x +64[/tex]

Step by step explanation:

[tex]\text{Given that, two roots are}~ -4~ \text{and}~ 4i.\\\\\text{Let,}\\\\~~~~~~~x = 4i\\\\\implies x^2 = 16i^2~~~~~~~;[\text{Square on both sides}]\\\\\implies x^2 = -16~~~~~~~~;[i^2 = -1]\\\\\implies x^2 +16 = 0\\\\\text{So,}~ x^2 +16~ \text{ is a factor of the 3 degree polynomial}.\\ \\ \text{The polynomial is ,}\\\\ f(x) = (x+4)(x^2 +16)\\\\~~~~~~~=x^3 +16x +4x^2 +64\\\\~~~~~~~=x^3 +4x^2 +16x +64[/tex]

Answer:

x ≈ 0.67

Step-by-step explanation:

[tex]15(3)^2x = 90\\\\15(9)x = 90\\\\135x = 90\\\\x = 90/135\\\\ x = 2/3\\\\x = 0.666...\\\\x = 0.67 \ (rounded \ to \ nearest \ hundredth )[/tex]

Answer:

36

Step-by-step explanation:

[tex]-2[-6(-4+7)]\\\\=(-2)(-6)(3)\\\\=12(3)\\\\=36[/tex]

Answer:

4

Step-by-step explanation:

x = 7 - 3

x = 4

Answer:

x = 4

Step-by-step explanation:

3+x = 7

Subtract 3 from each side

3+x-3 = 7-3

x = 4

2 + 7 = 9

6 + 4 = 10

2 + - 7 = -5

- 2 + 7 = 5

- 2 + - 7 = -9

6 + - 4 = 2

- 6 + 4 = - 2

- 6 + - 4 = -10

Answer:

see explanation

Step-by-step explanation:

PQRS is a cyclic quadrilateral.

the opposite angles in a cyclic quadrilateral sum to 180° then

∠ Q + ∠ S = 180 ( substitute values )

3x - 8 + 3x + 8 = 180

6x = 180 ( divide both sides by 6 )

x = 30

Then

∠ Q = 3x - 8 = 3(30) - 8 = 90 - 8 = 82°

∠ S = 3x + 8 = 3(30) + 8 = 90 + 8 = 98°

∠ P = x + 15 = 30 + 15 = 45°

∠ R = 180° - ∠ P = 180° - 45° = 135°

Answer:

[tex] \frac{1}{24} miles \\ [/tex]

Step-by-step explanation:

[tex] \frac{7}{8} - \frac{5}{6} = \frac{1}{24} [/tex]

Answer:

t would be steeper

Step-by-step explanation:

Answer:

16.67

Step-by-step explanation:

formula for a rectangular pyramid =

lengthxwidth= base area x height/3

2x5=10x5=50

50/3=16.67

Answer:

D1×D2/2sq

445×5= 225

225/2

=112.5

Answer:

See explanation

Step-by-step explanation:

Since she used only 1/2 of the recipe and the full recipe uses 1 cup, divide 1 by 2. The equation is 1/2=0.5.

For the lemon juice, the equation is 1/4 divided by 2. On the number line, make a dot at 1/4, and then travel backwards halfway. 1/4 divided by 2 is 1/8, or 0.125.

Hope this helps!

Step-by-step explanation:

In given, it's said that D is a midpoint, which mean DP == AD, DP is the same as DP because isosceles triangle properties said that two-length are equal when their base angle are equal. In a right triangle, the midpoint D, draws a line to the "90 degree" angle is always a isosceles

Answer:

Compound interest, or 'interest on interest', is calculated using the compound interest formula.

The formula for compound interest is A = P(1 + r/n)^nt, where P is the principal balance, r is the interest rate, n is the number of times interest is compounded per time period and t is the number of time periods.

Step-by-step explanation:

Answer:

dont now the answer but air math app will help you just take photo of problems

Answer:

4.08

Step-by-step explanation:

see the attachment for solution

hope it helps...!!!

Answer:

  $5.83

Step-by-step explanation:

The total amount of money is the sum of the values of the various denomination coins.

__

Let p, n, d, q represent the numbers of pennies, nickels, dimes, and quarters, respectively.

  p + n + d + q = 50 . . . . . . there are 50 coins in all

  16% × 50 = p . . . . . . . . . . 16% are pennies

  34% × 50 = d . . . . . . . . . . 34% are dimes

  n = p +3 . . . . . . . . . . . . . . there are 3 more nickels than dimes

The total value is ...

  v = 0.01p +0.05n +0.10d + 0.25q

__

The numbers of pennies and dimes are found easily:

  0.16 × 50 = p = 8

  0.34 × 50 = d = 17

Then the number of nickels can be found:

  n = p +3 = 8 +3 = 11

Finally, the number of quarters is found from the total number of coins:

  p + n + d + q = 50

  8 + 11 + 17 + q = 50 . . . substitute known values

  q = 14 . . . . . . . . . . . . . subtract 36 from both sides

The total value is then ...

  v = 0.01(8) +0.05(11) +0.10(17) +0.25(14) = 0.08 +0.55 +1.70 +3.50

  v = 5.83

The bag contains $5.83.

Answer: it might be 8

Step-by-step explanation: Write a distance formula:  x^2  - x^1    

substituted -3.8, 4.2 into  x^2  - x^1   and u will get 4.2 + 3.8 and u will calculate and it gets 8

I hope its right

You can see that both lines have the same slope showing that they are both parallel

These are equations that has a degree of 1. Given the equation of the lines

ab = 6x + 3y = 9

cd: 4x + 2y = 8

Write then in slope-intercept form

ab: y = -2x + 3

cd = y = -2x +8

You can see that both lines have the same slope showing that they are both parallel

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Answer:

They are parallel because they have the same slope of −2.

Step-by-step explanation:

Answer:

Surface area is 288π cm² (exact solution), which is approximately 904.78 cm² (if you use pi on calculator) or 904.32 cm² (if you use 3.14 for pi).

Step-by-step explanation:

Surface area of cylinder

[tex]SA = 2 \times \text{area of base} + \text{lateral area}\\SA = 2 \pi r^2 + 2 \pi r h[/tex]

SA ... surface area

r ... radius

h ... height of cylinder

Given:

d = 12 cm

h = 18 cm

Since diameter d is 12 cm, that means that radius is half of that, so 6 cm.

r = 6 cm

Insert in formula:

[tex]SA = 2 \pi r^2 + 2 \pi r h\\SA = 2 \pi (6)^2 + 2 \pi (6)(18)\\SA = 72 \pi + 216 \pi\\SA = 288 \pi \text{ cm}^2 \approx 904.78 \text{ cm}^2[/tex]

Answer:

[tex]\frac{26}{3}[/tex]=[tex]8.6...[/tex]=[tex]8\frac{2}{3}[/tex]

Step-by-step explanation:

You multiply 9 x 3=27

(leave 1/3 alone, it's already good)

you have a 27/3-1/3

(subtract those)

your answer is 26/3

(see answer above for decimal form and mixed fraction form)

The experimental probability that in a group of 4 students, at least one of them has brown eyes is 95%.

Experimental probability is a probability that is determined on the basis of a series of experiments. A random experiment is done and is repeated many times to determine their likelihood and each repetition is known as a trial. The experiment is conducted to find the chance of an event to occur or not to occur.

Here , Favorable Outcome = 19

          Total outcomes = 20

Probability = Favorable Outcome/ Total Outcome

                   = 19 / 20

                   = 95%

Thus, the experimental probability that in a group of 4 students, at least one of them has brown eyes is 95%.

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Answer:

Step-by-step explanation:

4x - 8 = 2x - 10

2x - 8 = -10

2x = -2

x = -1

y = 4(-1) - 8 = -4 -8 = -12

(-1, -12)