Answers
1. a root at 3i
2. a root at 2+4sqrt(3)
3. a root at -1/3
and that f(1)=-7520
Find polynomial f(x) as a product of its factors.
Given the three roots, we can conclude that the two other roots are the conjugates, namely -3i and 2-4sqrt(3).
Since
(x+3i)(x-3i)=x^2+9, and
(x-2+4sqrt(3))(x-2-4sqrt(3)=x^2-4x-44
We conclude that polynomial f(x) has the form
f(x)=a(x^2+9)(x^2-4x-44)(3x+1)
where a is a constant to be determined.
Knowing that f(1)=-7520, we can substitute x=1 in f(x) to get
-7520=a(1^2+9)(1^2-4(1)-44)(3+1)=a(10)(-47)(4)
=>
a=-7520/(-1880)=4
Therefore the polynomial f(x) is defined as
f(x)=4(x^2+9)(x^2-4x-44)(3x+1)
or
f(x)=4(3x+1)(x^2+9)(x^2-4x-44)
The root at 3i means that (x -3i)*(x +3i) = (x^2 +9) is a factor.
The root at 2 +4√3 means that (x -2 -4√3)*(x -2 +4√3) = (x^2 -4x -44) is a factor.
Evaluated at x=1, the product of these factors is (4)*(10)*(-47) = -1880, so there is an additional factor of -7520/-1880 = 4.
The intercept form of the polynomial is
.. y = 4(3x +1)*(x^2 +9)*(x^2 -4x -44)
Related Questions
The temperature of a cup of coffee t minutes after it has been poured is described by the function
C(t) = 20 + 60(0.95)^t .
a. How many minutes does it take for the coffee to cool to a temperature of d degrees?
b. About how fast is the temperature dropping when t = 1?
Answers
.. C(t) = d
.. d = 20 +60*0.95^t
.. (d -20)/60 = 0.95^t
.. log((d -20)/60) = t*log(0.95)
.. t = log((d -20)/60)/log(0.95) . . . . . . time to cool to d degrees (d > 20)
b) C'(t) = 60*0.95^t*ln(0.95)
.. C'(1) = 60*0.95*ln(0.95) ≈ -2.924 °C/min
Solve for x 16 = -2 -1
Answers
16x= -3
divide both sides of the equation by 16
x= -3/16
Alternative form= -0.1875
Susan bought a 2 kg bag of grapes on the way home she ate 125 g of the grapes how many grams of grapes that she have left
Answers
Select the scenarios that correctly represent the given graph.
Answers
The graph shown is an exponential graph. From the first point to the second, it increase from 100 to 150. This is a 50% increase. This pattern continues throughout the chart.
The first two choices give an increase of 50%. Or 1.5 times larger.
Answer:
The correct options are 1 and 2.
Step-by-step explanation:
From the given graph it is clear that it is a graph of an exponential function and the graph passes through the points (0,100) and (1,150).
The general form of an exponential function is
[tex]y=ab^x[/tex] .... (1)
Where, a is the initial value and b is growth rate.
The initial value is 100 because the y-intercept of the graph is 100.
The graph passes through (1,150). Put y=150, a=100 and x=1 in the above formula to find the value of b.
[tex]150=100b^1[/tex]
Divide both the sides by 100.
[tex]1.5=b[/tex]
The value of b is 1.5.
Substitute a=100 and b=1.5 in equation (1).
[tex]y=100(1.5)^x[/tex]
It means the value is 1.5 times larger in every unit of time.
It can be written as
[tex]y=100(1+0.5)^x[/tex]
It means the initial value increased by 50% per unit of time.
So, the required scenarios are
1. The amount of money in a bank account increased by 505 per year.
2. The amount of a substance is one and a half times larger after every minute.
Therefore the correct options are 1 and 2.
The pair of points is on the graph of an inverse variation. Find the missing value.
(1.6, 6) and (8, y)
0.83
30
0.03
1.2
WILL GIVE BRANLIEST ANSWER! HELP
Answers
6/5 = 1.2
The second point is (8, 1.2), corresponding to the 4th selection.
In the triangle below, which is equivalent to sinA?
Answers
Sin(α)=Opposite/Hypotenuse
2. Then, SinA is:
SinA=BC/AC
A is the angle in the vertice "A".
BC is the opposite leg to the the angle A.
AC is the hypotenuse of the triangle.
3. By definition, you have:
Cosα=Adjacent/Hypotenuse
4. Then, CosC is:
CosC=BC/AC
C is the angle at the vertice "C".
BC is the adjacent leg to the angle C.
AC is the hypotenuse.
5. Therefore, the answer is the last option:
The equivalent to SinA is CosC
sin A=BC/AC
and
cos C=BC/AC
therefore
sin A= cos C
the answer is cos C
Calculate the averages for each category for the three-month period. category jan. feb. mar. average wages $2,100.00 $2,300.00 $2,000.00 $ donations $210.00 $230.00 $200.00 $ mortgage $590.50 $590.50 $590.50 $ electricity $101.25 $89.75 $80.00 $ water $20.10 $22.30 $24.55 $ car loan $172.54 $172.54 $172.54 $ food $200.25 $220.70 $205.50 $ insurance $310.50 $0.00 $0.00 $ clothes $0.00 $0.00 $75.00 $ car license fee $0.00 $120.90 $0.00 $ newspaper $0.00 $0.00 $45.00 $ movies $18.00 $0.00 $36.00 $ new car $100.00 $100.00 $100.00 $ gasoline $80.00 $70.00 $75.00 $ car repairs $0.00 $119.00 $0.00 $ miscellaneous $64.50 $59.25 $72.50 $ savings $140.00 $140.00 $140.00 $
Answers
Jan Feb March
1)
wages $2,100.00 $2,300.00 $2,000.00
sum of the values = 2100 + 2300 + 2000 = 6400
number of values = 3
average = 6400/3 = $ 2133.33
2)
donations $210.00 $230.00 $200.00
average = (210 + 230 + 200)/3
= $ 213.33
3)
mortgage $590.50 $590.50 $590.50
average = ( 590.50 + 590.50 + 590.50)/3
= $ 590.50
4)
electricity $101.25 $89.75 $80.00
average = (101.25 + 89.75 + 80.00)/3
= 271/3
= $ 90.33
5)
water $20.10 $22.30 $24.55
average = (20.10 + 22.30 + 24.55)/3
= 66.95/3
= $ 22.32
6)
car loan $172.54 $172.54 $172.54
average = (172.54 + 172.54 + 172.54)/3
= $ 172.54
7)
food $200.25 $220.70 $205.50
average = (200.25 + 220.70 + 205.50)/3
= $ 208.82
8)
insurance $310.50 $0.00 $0.00
average = ( 310.50 + 0.00 + 0.00)/3
= $ 103.50
9)
clothes $0.00 $0.00 $75.00
average = (0.00 + 0.00 + 75.00)/3
= $ 25.00
10)
car license fee $0.00 $120.90 $0.00
average = (0.00 + 120.90 + 0.00)/3
= $ 40.30
11)
newspaper $0.00 $0.00 $45.00
average = (0.00 + 0.00 + 45.00)/3
= $ 15.00
12)movies $18.00 $0.00 $36.00
average = (18 + 0.00 + 36.0)/3
= $ 18.00
13)
new car $100.00 $100.00 $100.00
average = (100 + 100 + 100)/3
= $ 100.00
14)
gasoline $80.00 $70.00 $75.00
average = (80.00 + 70.00 + 75.00)
= $ 75.00
15)
car repairs $0.00 $119.00 $0.00
average = (0.00 + 119.0 + 0)/3
= $ 39.67
16)
miscellaneous $64.50 $59.25 $72.50
average = (64.50 + 59.25 + 72.50)/3
= 196.25/3
= $ 65.42
17)
savings $140.00 $140.00 $140.00
average = ( 140 + 140 + 140)/3
= $ 140.00
You can upgrade lighting at your factory to LED bulbs that cost $6.95 each and last an average of 5 years. It costs $3 in labor to change a bulb. Over a 10-year period, about how much will it cost per year to install LED bulbs in 100 lamps and change the bulbs when they burn out?
Answers
.. $6.95 +$3 = $9.95
100 bulbs and 10 years totals (100*10) = 1000 bulb·years. For the price of $9.95, you purchase 5 bulb·years, so the total cost for 1000 bulb·years is
.. (1000 b·y)*($9.95/(5 b·y)) = $1990
Over a 10-year period, installation and replacement of 200 bulbs in 100 lamps will cost $1990. Then the per-year cost is ...
.. $1990/(10 yr) = $199/yr
Answer: $200.00
Step-by-step explanation:
$6.95+ $3.00= $19.90
$19.90 * 100= $1,990
$1,990 / $9.95 = $200.00
take the orginal cost 6.95 plus the $3 labor which equals 19.90 then you multiply by the number of bulbs 100 which equals 1990 then divide 1990 by the cost of bulbs 9.95 which equals 200!
A city was founded at the beginning of 1990 with a population of 55,000, and since then, the growth in its population has been exponential, increasing at x percent per year. If the city's population at the beginning of 2000 was 108,193, at what percent per year to the nearest percent is the city's population increasing?
Answers
.. 55000*(1 +x/100)^10 = 108193
.. (1 +x/100)^10 = 108193/55000
.. 1 +x/100 = (108193/55000)^(1/10)
.. x = 100*((108193/55000)^(1/10) -1) ≈ 7.00
but I find it just as fast to have a graphing calculator do it.
The city's population is increasing 7% per year.
Multiple Choice
For questions 1–7, choose the word or phrase that best defines the italicized word.
1. "Only reapers reaping early/In among the bearded barley…" (1 point)
people who toil
people who obey
people who
Answers
Answer:
people who toil
Step-by-step explanation:
The above lines have been taken from the poem "The Lady of Shalott" written by Alfred, Lord Tennyson. The poem is about the mysterious Lady of Shalott who lives on the Island of Shalott. People who wanted to see her have never got the opportunity to see her. On the other hand, the people who work in the fields have listened to her singing an eerie song. The song which the lady sings echoes all over the river Camelot.
Which out of these 2 "graphs" represent a proportional relationship? PLEASE HELP
x f(x)
–1 –2
0 0
1 2
4 8
x f(x)
–2 2
0 4
2 6
3 7
Answers
How do I find x in this triangle?
Answers
The value of a textbook is $120 and decreases at a rate of 12% per year. Write a function to model the situation and then find the value of the textbook after 9 years.
Answers
After the first deduction, the book now costs $105.60.
It decreases $14.40 per year.
Do this for 8 more years.
This should give you what you need to form the needed function modeling the situation.
To model the depreciation of a textbook initially worth $120 with a 12% annual decrease, use the function V(t) = 120 x (1 - 0.12)^t. After 9 years, the textbook's value will be approximately $38.24.
The student asks for a function to model the depreciation of a textbook and the value of the textbook after 9 years. The textbook costs $120 initially and depreciates at 12% annually. To model this, we can use an exponential decay function, which is generally of the form V(t) = V_0 imes (1 - r)^t, where V(t) is the value after time t, V_0 is the initial value, r is the rate of decay, and t is time in years.
Plugging in the given values:
V_0 = $120
r = 0.12 (12% as a decimal)
t = 9 years
The function becomes V(t) = 120 imes (1 - 0.12)^t. Now, we find the value of the textbook after 9 years by substituting t with 9:
V(9) = 120 x (1 - 0.12)⁹
Calculate:
V(9) = 120 x (1 - 0.12)⁹
V(9) = 120 x 0.88^9
V(9) =120 x 0.318630...\approx $38.24
Thus, the value of the textbook after 9 years will be approximately $38.24.
What's the recursive formula?? help!
Answers
Find an explicit formula for the arithmetic sequence 37, 74, 111, 148
Answers
a₂ = 74
d = a₂ - a₁ = 74 - 37 = 37
a(n) = 37 + 37(n-1)
Let us observe the given series 37,74,111,148
Here, the first term (a) = 37
Common difference(d) = 74-37 = 37
Hence, the given series is in Arithmetic progression.
Hence the nth term of the series is given by:
[tex] a_{n}=a+(n-1)d [/tex]
Substituting the values of 'a' and 'd'.
[tex] a_{n}=37+37(n-1) [/tex]
Therefore, it is the required explicit formula.
Carl borroowed $2,500 for six months at an annual interest rate
of 11%. How much must he repay at the end of six months?
a. $275
b. $2,637.50
c. $87.50
d. $2,762.50
Answers
I hope I helped!
Please answer. Confusing me. 20 points.
Answers
Read the word problem. Cate has 10 dolls. Shelayna has 3 times as many dolls as Cate. How many more dolls does Shelayna have compared to Cate? What operations are needed to solve this word problem, and in what order should they be used? addition and subtraction, in that order
addition and division, in that order
division and subtraction, in that order
multiplication and subtraction, in that order
Answers
Therefore to find how many dolls Shelayna has we have to first multiply 10 * 3
Thats 30 dolls
Then we have to find how much more Shelayna has compared to Cate, for this we need to find the difference between the number of dolls they both have.
To find differences we have to subtract the number of dolls Cate has from the number of dolls Shelayna has, thats 30 - 10 = 20
Therefore the operations are multiplication and subtraction in order
Answer:
multiplication and subtraction in order
Step-by-step explanation:
^ please help me____________
Answers
The two numbers that satisfy the condition quadratics have is
1. They must add to 6
2, They must multiply to - 16
All quadratics have to satisfy these two things.
The only numbers that will are 8 and - 2
So the factors are
(x - 8)(x + 2) = 0
x - 8 = 0
x = 8
and
x + 2 = 0
x = - 2
Sean Wrona is the fastest typist in the world and can type an average of about 500 words in three minutes. What is Sean’s average typing speed in words per minute, WPM, to the nearest whole number? *** Answer with number only***'
Answers
The triangles are similar. What is the value of x?
Answers
Answer:
The value of x is [tex]16\ units[/tex]
Step-by-step explanation:
we know that
If two figures are similar , then the ratio of its corresponding sides is equal
In this problem
[tex]\frac{x}{4}=\frac{12}{3}=\frac{20}{5}[/tex]
therefore
[tex]\frac{x}{4}=4[/tex]
[tex]x=16\ units[/tex]
What is the value of n?
Enter your answer in the box.
n =
Answers
Answer:
The value of n is 4.
Step-by-step explanation:
Theorem of intersection of two cords : If two cords AB and CD intersecting at point P, then
[tex]|AP|\times |PB|=|CP|\times |PD|[/tex]
Using the theorem of intersection of two cords, we get
[tex](n-2)\times 16=8\times (n+8)[/tex]
Use distribution property,
[tex]16n+32=8n+64[/tex]
[tex]16n-8n=64-32[/tex]
[tex]8n=32[/tex]
Divide both sides by 8.
[tex]n=\frac{32}{8}[/tex]
[tex]n=4[/tex]
Therefore the value of n is 4.
Two one-step equations
Two equations that contains fractions
One equation with distributive property
One equation with decimals
One real-world problem that is solved by an equation
these equations must be for pre-algerbra
Answers
7x = 21
B. b - 2/3 = 5/6
1/5m = 20
C. 45 = 3(x + 1)
D. 22.5 = 1.5x
E. Mark has $20 in his bank account. He saves $5 each week. How much money does Mark have in his account after 9 weeks? Use w to represent the number of weeks and t to represent the total amount Mark has after any number of weeks.
Answer:
One step equations
3 + x = 5 ___x = 2
Y + 14 = 20____x = 6
Equations with Fractions
4/5q = 32___q = 40
3/4x = 6___x = 8
Equation with decimals
3.2x – 2.7 = 1.2x___x = 1.35
Equation with distributive Property
7x + 3 (x + 1) = 10x + 3
Infinitely Many solutions
Real World Problem
Monica knows that she should spend no more than 1/3 of her income on her rent. She really wants to move into a cool condo downtown and has found a great place where the rent is $2,300 per month. Based on that, how much should Monica earn each month to afford the rent on her cool new place?
Hope This Helps!
Logan can paint a room in 12 hours. Marcus can paint the same room in 16 hours. How long does it take for both Logan and Marcus to paint the room if they are working together?
Answers
The answer should be "4".
Reason:
Since Logan takes 12 hours and Marcus takes 16 i just subtract 16 and 12.
16-14=4
=4 hours
If you need anymore help feel free to ask me!
Hope this helps!
~Nonportrit
Of the 50 shapes in a set, 20 of them are squares. In a circle graph, what would be the measure of the central angle for the section that represents squares?
A. 40 degrees
B. 108 degrees
C. 126 degrees
D.144 degrees
Answers
Given the equation 2(3x − 4) = 5x + 6, solve for the variable. Explain each step and justify your process. Charlie solved a similar equation below. Is Charlie's solution correct? Explain why or why not.
Answers
In order to solve this problem, we need to simplify.
2 * 3x - 2 * 4 = 5x + 6
6x - 8 = 5x + 6
-8 = -x + 6
8 = x - 6
14 = x
x = 14
Charlie's answer was not stated, so I don't know whether he was correct or not ;)
Hope this helps!
In this fulcrum, the weights are perfectly balanced. How far must the fulcrum be located from the 60 pound weight if the bar is 24 feet long?
A) 9
B) 15
C) 12
Answers
Answer:
15 feet
Step-by-step explanation:
Given : In this fulcrum, the weights are perfectly balanced.
To Find: How far must the fulcrum be located from the 60 pound weight if the bar is 24 feet long?
Solution:
To make the weights perfectly balanced torque must be equal
[tex]Torque = Force \times d[/tex]
Where [tex]Force = Mass \times Acceleration[/tex]
Let a be the acceleration
d= Distance between the pivot and the acting point.
Let x be the distance from 60 pound weight where fulcrum is located
Since we are given that Length of bar = 24 feet.
So,Distance of fulcrum from 100 pound weight = 24-x
Now torque for 60 pound weight :
[tex]Torque = 60a \times x[/tex]
Now torque for 100 pound weight.
[tex]Torque =100a \times (24-x)[/tex]
Now to maintain the equilibrium i.e. To make the weights perfectly balanced
[tex]60a \times x =100a \times (24-x)[/tex]
[tex]60x = 2400-100x[/tex]
[tex]160x = 2400[/tex]
[tex]x = \frac{2400}{160}[/tex]
[tex]x = 15[/tex]
Hence The fulcrum must be located 15 feet from the 60 pound weight if the bar is 24 feet long.
Answer:
15
Step-by-step explanation:
PLEASE HELP !!! 15 POINTS !! PLUS BRAINLIEST !!
The graph shows the shape of an arched bridge cable. What does the y-intercept represent?
The total length of the cable is 15 m.
The left end of the cable is 15 m above the roadway.
The minimum distance between the cable and roadway is 15 m.
The total length of the road is 15 m.
Answers
Answer:
The left end of the cable is 15 m above the roadway
Step-by-step explanation:
Let
x----> the horizontal distance from left end of cable (m)
y----> the height above roadway (m)
we know that
The y-intercept is the value of y when the value of x is equal to zero
In this problem
The y-intercept is the height above roadway (m) when the distance from end of cable is equal to zero
so
The y-intercept is the point (0,15)
so
The left end of the cable is 15 m above the roadway
The left end of the cable is 15 m above the roadway]
Step-by-step explanation:
what does a^2+b^2=C^2 mean
Answers
The equation a² + b² = c² represents the Pythagorean theorem, which describes the relationship between the sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. For example, in a triangle with legs of 3 and 4 units, the hypotenuse would be 5 units.
The equation a² + b² = c² represents the Pythagorean theorem in mathematics. This theorem is used to describe the relationship between the lengths of the sides of a right triangle.
Pythagorean Theorem
In a right triangle:
a and b are the lengths of the legs (the two sides that form the right angle).
c is the length of the hypotenuse (the side opposite the right angle).
The theorem states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides:
a² + b² = c²
This can also be expressed as c = √(a² + b²).
For example, if one leg of the triangle is 3 units and the other leg is 4 units, you can find the hypotenuse as follows:
3² + 4² = c²9 + 16 = 25c = √25 = 5Question:
what does a^2+b^2=C^2 formula mean ?
A small company has 20,000 shares. Anastasia owns 400 of these shares. The company decides to split its shares. What is Anastasia's ownership percentage after the split?
Answers
Answer:
2%
Step-by-step explanation:
A P E X