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3 years ago

(30 POINTS HELP) What is the complete factorization of the polynomial function over the set of complex numbers?

Mathematics

1 answer:

earnstyle [38]3 years ago

5 0

Answer:

$f(x) =( {x} - 4i)(x + 4i)(x + 3)$

Step-by-step explanation:

The given polynomial is

$f(x) = {x}^{3} + 3 {x}^{2} + 16x + 48$

Let us factor by grouping:

$f(x) = {x}^{2} (x + 3) + 16(x + 3)$

We factor further to get:

$f(x) = ( {x}^{2} + 16)(x + 3)$

We need to get the quadratic term factored.

$f(x) =( {x}^{2} - ( - {4}^{2} ))(x + 3)$

$f(x) =( {x}^{2} - ( {4i)}^{2} )(x + 3)$

We apply difference of two squares to get:

$f(x) =( {x} - 4i)(x + 4i)(x + 3)$

This is the completely factored form over the complex numbers.

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What is 4/ 6.4 simplified as?

Mashcka [7]

Answer:

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Step-by-step explanation:

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3 years ago

Read 2 more answers

A card is being drawn from a standard deck of playing cards. Find each probability.

EastWind [94]

These are 6 questions and 6 answers.

To find each probability we will use the definition of probability:

Probability = number of positive outcomes / number of total possible outcomes

1) P(Jack or ten)


Answer: 2/13 ≈ 0.12


Justification:

i) Positive outcomes: A standard deck of cards has 4 jacks and 4 tens, then those are 4 + 4 = 8 different positive outcomes.

ii) Possible outcomes: a standard deck of cards has 52 different cards, so, that is a total of 52 different possible outcomes

iii) Probability, P

P = number of positive outcomes / number of total possible outcomes

P = 8 / 52 = 2/13 ≈ 0.15

 2.P(red or black)


Answer: 1

Justification:

i) Positive outcomes

Half of the cards are red and half of the cards are black, so they both add for the total of the cards = 52

ii) Possible outcomes: 52 cards

iii) Probaility, P

P = number of positive outcomes / number of total possible outcomes

P = 52 / 52 = 1

 3.P(queen or club)


Answer: 4/13 ≈ 0.31

Justification:

i) Positive outcomes

There are 4 Queens.

There are 1/4 of 52 clubs = 1/4 × 52 = 13 clubs.

But you cannot add all of them, because one club is the Quenn of Clubs.

Then, the total number of different Queens and clubs is 4 + 13 - 1 = 16

ii) Possible outcomes: 52 different cards

iii) Probaility, P

P = number of positive outcomes / number of total possible outcomes

P = 16 /52 = 4 / 13 ≈ 0.31

 4.P(red or ace)


Answer: 7 / 13 ≈ 0.54

Justification:

i) Positive outcomes

Half of the cards are red: 26

There are 4 aces.

Since 2 aces are red, the number of different red and aces cards is: 26 + 4 - 2 = 28

ii) Possible outcomes: 52 different outcomes

iii) Probaility, P

P = number of positive outcomes / number of total possible outcomes

P = 28 / 52 = 7 / 13 ≈ 0.54

 5.P(diamond or black)


Answer: 1/2 = 0.5

Justification:

i) Positive outcomes

There are 52 / 4 = 13 diamonds

There are 26 black cards.

All the diamonds are black cards.

Then, the number of different diamond or black cards is 13 + 26 - 13 = 26

ii) Possible outcomes: 52 different cards.

iii) Probaility, P

P = number of positive outcomes / number of total possible outcomes

P = 26 / 52 = 1/2 = 0.5

6.P(face card or spade)

Answer: 11/26 ≈ 0.42

Justification:

i) Positive outcomes

Face cards are jacks, queens and kings. That is 3 × 4 = 12 different cards.

The spades are 13 cards.

Since, 3 of the faces are spade cards, the number of different cards of those types are 12 + 13 - 3 = 22

ii) Possible outcomes: 52 different cards

iii) Probaility, P

P = number of positive outcomes / number of total possible outcomes

P = 22 / 52 = 11 / 26 ≈ 0.42

4 0

3 years ago

kvv77 [185]

Answer:

so the right answer is of option D

Step-by-step explanation:

Have a great day and hope this helps!!!

7 0

3 years ago

there are 9000 four digit numbers. 4536 of the four digit numbers have no repeating digits. How many four digit numbers have at

Shtirlitz [24]

This is subtraction:

9,000-4,536=4,464

5 0

3 years ago

A 24-centimeter by 119-centimeter piece of cardboard is used to make an open-top box by removing a square from each corner of th

satela [25.4K]

Answer:

The size square removed from each corner = 32.15 cm²

Step-by-step explanation:

The volume of the box = Length * Breadth * Height

Let r be the size removed from each corner

Note that at maximum volume, $\frac{dV}{dr} = 0$

The original length of the cardboard is 119 cm, if you remove a size of r (This typically will be the height of the box) from the corner, since there are two corners corresponding to the length of the box, the length of the box will be:

Length, L = 119 - 2r

Similarly for the breadth, B = 24 - 2r

And the height as stated earlier, H = r

Volume, V = L*B*H

V = (119-2r)(24-2r)r

V = r(2856 - 238r - 48r + 4r²)

V = 4r³ - 286r² + 2856r

At maximum volume dV/dr = 0

dV/dr = 12r² - 572r + 2856

12r² - 572r + 2856 = 0

By solving the quadratic equation above for the value of r:

r = 5.67 or 42

r cannot be 42 because the size removed from the corner of the cardboard cannot be more than the width of the cardboard.

Note that the area of a square is r²

Therefore, the size square removed from each corner = 5.67² = 32.15 cm²

5 0

3 years ago