Given that x² -x -2 is a factor of x³ +3 x² +ax +b, calculate the values of a and b and hence find the remaining factor.

Answers

Answer 1

Answer:

[tex]a = -6, b = - 8[/tex]

Remaining factor = [tex]x+4[/tex]

Step-by-step explanation:

Given that:

Two polynomials:

[tex]x^{2} -x-2[/tex] and [tex]x^3 +3 x^2 +ax +b[/tex]

[tex]x^{2} -x-2[/tex] is a factor of [tex]x^3 +3 x^2 +ax +b[/tex].

To find:

The values of [tex]a[/tex] and [tex]b[/tex] from the cubic equation and the remaining factor.

Solution:

Let us first of all, factorize [tex]x^{2} -x-2[/tex].

[tex]x^{2} -2x+x-2=0\\\Rightarrow x(x-2)+1(x-2)=0\\\Rightarrow (x+1)(x-2)=0\\\Rightarrow x = -1, 2[/tex]

There are two factors of the given quadratic equation.

These two factors must also be factors of the cubic equation as well.

Putting [tex]x = 2[/tex]:

[tex]2^3 +3 \times 2^2 +a\times 2 +b =0\\\Rightarrow 2a+b=-20 ..... (1)[/tex]

Putting [tex]x = -1[/tex]:

[tex](-1)^3 +3 \times (-1)^2 +a\times (-1) +b =0\\\Rightarrow -a+b=-2 ..... (2)[/tex]

Subtracting (2) from (1):

[tex]3a = -18\\\Rightarrow a = -6[/tex]

From equation (1):

[tex]2\times (-6) + b=-20\\\Rightarrow b = -8[/tex]

Putting the values of [tex]a[/tex] and [tex]b[/tex] in the cubic equation, we get:

[tex]x^3 +3 x^2 -6x -8[/tex]

Dividing the cubic equation with quadratic, we get:

[tex]\dfrac{x^3 +3 x^2 -6x -8}{x^{2} -x-2} = x+4[/tex]


Related Questions

PLEASE HELPPP IDEK WHEN WE LEARNED THISSS

Answers

Answer:

1 in 52 I THINK DONT BE MAD IF ITS WRONG :(

Step-by-step explanation:

Because there are 52 cards in the deck and you need to pull out 1 card so its 1 in 52 chance

1 of 13 because there are 4 different sets of numbers, letters, shapes, and colors for each.

What is 135 less than the product of 57 and 219?

Answers

Answer: something in the negatives

Step-by-step explanation:

Please answer correctly will mark brainliest and follow thank you ;)

Answers

Answer:

76m

Step-by-step explanation:

Solve perimeter of both rectangles and add

Answer:

238 meters

Step-by-step explanation:

Perimeter? The equation is P = 2 ( l * w )

So lets cut this irregular shape into 2 rectangles.

I added a picture to show you.

So on rectangle is 7 by 11 and the other is 7 by 6.

So lets find the perimeter of the larger rectangle first.

P = 2 ( 11 * 7 )

P = 2 ( 77 )

P = 154

Now the smaller one

P = 2 ( 7 * 6 )

P = 2 ( 42 )

P = 84

Now to find the full perimeter, add both perimeters together

P = 154 + 84

P = 238

238 meters

Don't be afraid to reach out with further questions! Hope this helps!

What is the equation

Answers

Answer:

x=15

Step-by-step explanation:

6+9=15

so x=15

UGHHHHHH HELPPP!!!!!!!! The sum of a number and 3 more than twice the number is less than 36. Find the number! ANSWER QUICKLYYYY

Answers

Step-by-step explanation:

Read the poem. excerpt from “Spring” by Christina Rossetti Frost-locked all the winter, Seeds, and roots, and stones of fruits, What shall make their sap ascend That they may put forth shoots? Tips of tender green, Leaf, or blade, or sheath; Telling of the hidden life That breaks forth underneath, Life nursed in its grave by Death. Which line from the poem contributes most to a hopeful tone? "Telling of the hidden life" "Frost-locked all the winter," "Life nursed in its grave by Death." "What shall make their sap ascend"

first question is Bitcoin is a virtual currency used to buy things online. An investment of $5125 in bitcoins grew 48 times. What is the total value of the investment?

And the second question is The cost of each projector in an office is $9841. Find the cost of 210 such projectors.
$
tell me the answer A S A P plz

Answers

[tex]\huge\mathbb\purple{AnSwEr}[/tex]

[tex]\huge❤\bold\red{Hello}❤[/tex]

Hey mate 1. $246,000 2. $20,66,610 is your Answer

[tex]My \: pleasure \: to \: help \: you \: anytime.[/tex]

[tex]Mark \: me \: as \: brainliest .[/tex]

[tex] \huge \mathscr \pink \star\underline {Follow \: me } \red \star[/tex]

There are 12 squares and 20 triangles. What is the simplest ratio of squares to triangles?

Answers

Answer: 12:20 but if simplified then it is 3:5! This is the simplest ratio of square to triangles Explanation: I found the "GCF" (Greatest Common Factor) which was 4 and I got 3 for 12/4 and 5 for 20/4

its not math but I still need helppppp​

Answers

Answer:

T

Step-by-step explanation:

how are angles of an equilateral triangle related?

Answers

All are 60 degrees because of equal lengths and equal angles
Since 180 ° in triangle means 180/3 =60

Hope this helps!

Answer:

If the triangle is equilateral, then the triangle is also equiangular.

Step-by-step explanation:

(10-3)(-2)

giving brainlist out

Answers

Answer:

i think the answers 5

Step-by-step explanation:

Answer:

-14

Step-by-step explanation:

(10-3)(-2)

(7)(-2)

-14

evaluating reports , help me please ...

Answers

Well I think it’s b

27
Karim (22 years) tells his brother Ziad (8 years).
«l haven't given you a gift for your birthday, but I'll make it up to you. The year when my age will be
triple yours. I'll give you a wonderful gift».
After making his calculations, Ziad tells his brother :
It will cost you nothing». Explain why.
(solve it in an equation of 1 unknown)​

Answers

Answer:

Karim will never have triple age as his brother, it has happened already 1 year ago, thus he won't give him a gift

Step-by-step explanation:

Equations

Let's make:

x = the number of years from now

Karim now is 22 and Ziad is 8. x years from now Karim will be

22 + x years old and Ziad will be

8 + x years old.

Karim will give Ziad a wonderful gift when:

22 + x = 3(8 + x)

22 + x = 24 + 3x

Solving:

x = -1

This means that Karim will never have triple age as his brother, it has happened already 1 year ago, thus he won't give him a gift

I need help will give crown

Answers

Answer:

$15 per hour and he worked 5 hours yesterday.

Step-by-step explanation:

Since we know that the number of hours he works and know how much he got paid we need to look for how much he makes each hour.

Set up the equation like this.

7x=105

Divide by 7 to isolate the x

x=15

That means he makes 15 dollars per hour. For the 2nd part, we are looking for how many hours he worked. We know that he gets 15 dollars per hour so we can set up the equation like this.

15x=75

Divide by 15 to isolate the x

x=5

That means he worked 5 hours yesterday.

P.S. give me crown

Answer:

 He worked approximately 4-5 hours because when he worked 7 hours a day he earned 105 dollars that means he made 15 dollars an hour so he only worked 4-5  hours that day and he made 75 dollars.  

Step-by-step explanation:

This is a short answer so u dont have to write that much.

Fill in the missing fractions in the blanks. (You don't need to simplify.) 50.891 = 50 + 8/10 + Source 51-5 USM2 THAN YO UNA CA bo Anu​

Answers

Answer:

Sorry no idea

Step-by-step explanation:

Write seventeen thousand and one hundred six thousandths in standard form

Answers

Answer: 17,000.106.

Step-by-step explanation: hope this helps

Find the x-intercept and y-intercept of the line.
2x+y=8
y-intercept:
x-intercept:

Answers

Answer:

The x intercept is 4 (by substituting y as zero we get x intercept)

And the y intercept is 8 (by substituting x as zero we get y intercept)

Take (4, 0) and take (8, 0) to graph the line

You're welcome!

Answer:

x = 4 y = 8 graph (4,0) and (8,0)

Addilyn received 300 of the most popular beads for her birthday.
There are 30 beads per package. How many packages did she receive?


1. a = 300 + 30

2. a = 300 x 30

3. a = 300 divided by 30

4. a = 300 - 30

Answers

The correct answer is 3: a= 300 divided by 30

Answer:

3. a=300/30

Step-by-step explanation:

you want to figure out how many packages she received, and there are 30 beads in each package. so, you divide the total amount of beads (300) by the amount in each package to determine how many packages there are. 300/30=10 packages of beads

What is the sum?
8+(-12)
-20
24
4
20

Answers

Answer:

-4

Step-by-step explanation:

adding a negative is the same as subtracting a positive

xoxo, your highness...

Answer: 24

8 + (-12) = -4

-4 + (-20) = -24

-24 + 24 = 0

0 + 4 = 4

20 + 4 = 24

Write a fraction equal to 1 with a numerator of 18

Answers

Answer:

18/18

Step-by-step explanation:

18 divided by 18 is one

if lines intersect, does that make it, one solution, no solution, or infinite solution?​

Answers

Answer:

one solution

Step-by-step explanation:

No solution is when they don't intersect at all. Infinite solutions is if the two lines are on top of each other.

Answer:

one solution

Step-by-step explanation:

if they intersect, they become one solution, if they don't intersect, than there are no solutions. If the graphs of the equations are the same, there are infinite solutions.

plz help me, thank you

A bacteria culture starts with 40 bacteria and the number doubles every 20 mins.
write an equation of an exponential function that models the described situation.

plz show me the working.

Answers

Answer:

[tex]P=40(1.03526)^{t}[/tex]

Step-by-step explanation:

Exponential Growth

The natural growth of some magnitudes can be modeled by the equation:

[tex]P=P_o(1+r)^{t}[/tex]

Where P is the actual amount of the magnitude, Po is its initial amount, r is the growth rate and t is the time.

The initial number of bacteria is Po=40 and it doubles (P=2Po) at t=20 min. With that point we can find the value of r:

[tex]2P_o=P_o(1+r)^{20}[/tex]

Simplifying:

[tex](1+r)^{20}=2[/tex]

Solving for 1+r:

[tex]1+r=\sqrt[20]{2}[/tex]

[tex]1+r=1.03526[/tex]

The exponential function that models the situation is:

[tex]\mathbf{P=40(1.03526)^{t}}[/tex]

Select the correct answer from the each drop-down menu. if 0

Answers

Answer:

You didn't provide a image of the problem how am I suppose to help you

Bill built a rectangular pen in his yard that measures feet by 6 feet. He wants to make another pen that is similar but not congruent to the one he built. Which these rectangles is similar?

Answers

You have to include a picture for us to see what’s the choices

3. Jason earns $400 per week plus a 3% commission on his total sales. If he sold $675 in total sales, how much did he earn this week?​

Answers

Answer:

$420.25

Step-by-step explanation:

Total earnings of the week

[tex] = 400 + 3 \% \: of \:675 \\ \\ = 400 + 0.03 \times 675 \\ \\ = 400 + 20.25 \\ \\ = \$ 420.25[/tex]

Today only, a suit is being sold at a 15% discount. The sale price is $391.
What was the price yesterday?

Answers

Answer:

$460.

Step-by-step explanation:

[tex]Percentage\ change = (\frac{Original\ Price\ - Discounted\ Price}{Original\ Price})100\\\\15=(\frac{x-391}{x})100\\\\15x=(x-391)100 \\\\15x/100=x-391\\0.15x=x-391\\391=x-0.15x\\391=0.85x\\391/0.85=x\\460=x\\[/tex]

So the price yesterday was $460

Sandy plays a matching game in order to advance to the next level, she must score at least 10 points. She scores 15 points for correct matches and -6 points for incorrect matches.

A.Write an addition expression to represent the amount of points Sandy earned.

B.Did Sandy make it to the next level? Explain

Answers

Answer:

A) Points earned = 15 - 6

B) Sandy didn't make it to the next level since her score was 9 and she needed a minimum of 10

Step-by-step explanation:

To advance to the next level in her game, Sandy must score at least 10 points.

She scores 15 points for correct matches

She scores -6 points for incorrect matches.

A. The amount of points earned by Sandy in the game is the sum of both scores:

points earned = 15 - 6

points earned = 9

B. Unfortunately, Sandy didn't make it to the next level since her score was 9 and she needed a minimum of 10

Can somebody please solve these for me I don't wana do it (thats why i am using so many points)

Answers

Down                            

6. 4,011                          

10. -94

11. 26

13. 178

15. -8

16. 6,861

18. 3,482

20. 385

22. 788

24. -5

25. 44

Across

1. -248

5. -9

7. 4,923

8. 90

9. 67

10. -912

12. 119

14. 16

15. -6

17. 743

19. 8,838

21. 47

23. 68

24. -884

26. 15

27. 7,284

Which of the following is true about similar figures?

All similar figures are congruent.


If two figures are not similar, they can still be congruent.


All congruent figures are similar.


If two figures are not congruent, they can't be similar.

Answers

Answer:

All congruent figures are similar

Step-by-step explanation:

Congruent means "same" in mathematics and you have evidence to prove it. You can look at two puppies with the exact same fur color and be like "yeah they're the same". If they are the same size and shape they're congruent. This is why the AAA postulate can't work because triangles can have the same angles yet be different sizes. If figures are not congruent, they can't be similar and figures that aren't similar can't be congruent either.

All similar figures are congruent is true about similar figures

What is congruency?

We know two similar planer figures are congruent when we have sides or angles or both that are the same as the corresponding sides or angles or both.

In mathematics, congruent means identical, and you can cite examples to support your claim.

just by looking at them. They are congruent if their dimensions match. Triangles can have the same angles but differ in size, which makes the AAA postulate invalid.

If two figures are not similar, they cannot be congruent, therefore similarly unrelated figures cannot be congruent.

learn more about congruency here :

https://brainly.com/question/7888063

#SPJ2

Christine purchased a prepaid phone card for $20. Long distance calls cost 18 cents a minute using this card. Christine used her card only once to make a long distance call. If the remaining credit on her card is $15.68, how many minutes did her call last?

Answers

Answer:

4 minutes and 32 seconds

Step-by-step explanation:

card $20.00-money left over $15.68            $20.00-$15.68  4.32

A car is 180 inches long. A truck is 75% longer than the car. How long is the truck?

Answers

Answer:

315 inches

Step-by-step explanation:

180 + 180 * 075 =

180 + 135 = 315

Answer:

The truck is 315 inches long.

Step-by-step explanation:

Other Questions
In what ways do you follow a healthy diet? -2.5f + 0.2f - 13 - 8 To find the quotient of 3 divided by one-sixth, multiply 3 byOne-sixth.Three-sixths.3.6. plsssss helpppp asap Which part of the carbon cycle occurs when plants, trees, or fossil fuels are burned?A. Transpiration B. Oxidation C. Photosynthesis D. Combustion Evaluate: 12 divided by y+9d if y = 5 and d = 7 There are 143 students, 6 teachers, and 8 chaperones going to the theater. To reserve their seats, they need to reserve entire rows. Each row has 9 seats.How many rows must they reserve? Adding two negative numbers together always results in a negative number." Is this statement True or False? Demonstrative Adjectives, Help Can u guys PLEASE ANSWER this question 2 ASAP One of the most recognizable features of Kabuki theater is: *the use of elaborate background setsthe polychromed wooden masks worn by the actors Owomen play all of the roles in a playthe bold makeup worn by the actors What are the five main ideas of Wilson's 14 points? How well did Wilson's 14 points do in the Treaty of Versailles? What will be the ultimate outcome of the treaty? I need help Plzzzzzzzzz brooke plays on her schools softball team. Last season she was at bat 500 times and made a safe hit 28% of the time. How is 28% expressed as a fraction PLEASE HELP please? Select the correct answer from each drop-down menu.What are the characteristics of Anorexia and Bulimia?An anorexic patient loses weight because ofA person with Bulimia suffers from 3. What is the energy of a photon whose frequency is 5.2 x 1015 Hz? Use the equation: E = hxvh= 6.626 10-34 J xs4. What is the frequency of radiation whose wavelength is 2.40 x 105 cm? v=c+XC= 3 x 108m/s graph the rational function f(x)=(3x-1)/(x-5) According to the lecture, which of the following is not a reason men communicate?a.to share informationb.to entertain an audiencec.to accomplish a taskd.all of these are not reasons men communicate Pls help me on this Ethnic groups share a common ______________while religious groups share a common_________________ .