Use the multinomial theorem to compute some polynomial expansions:
(a + b + c)² = a² + b² + c² + 2 (ab + ac + bc)
(a + b + c)³ = a³ + b³ + c³
… … … … … … + 3 (a²b + a²c + ab² + b²c + ac² + bc²)
… … … … … … + 6 abc
(a + b + c)⁴ = a⁴ + b⁴ + c⁴
… … … … … … + 4 (a³b + a³c + ab³ + b³c + ac³ + bc³)
… … … … … … + 6 (a²b² + a²c² + b²c²)
… … … … … … + 12 (a²bc + ab²c + abc²)
Now, given that
a + b + c = 4
a² + b² + c² = 10
it follows that
ab + ac + bc = (4² - 10)/2 = 3
We use this result to simplify the extra terms in the 3rd-degree expansion.
a²b + a²c + ab² + b²c + ac² + bc²
= a (ab + ac) + b (ab + bc) + c (ac + bc)
= a (ab + ac + bc) + b (ab + ac + bc) + c (ab + ac + bc) - 3abc
… … I underline the terms that are added and subtracted … …
= (a + b + c) (ab + ac + bc) - 3abc
= 4 • 3 - 3abc
= 12 - 3abc
Given that
a³ + b³ + c³ = 22
this tells us that
4³ = 22 + 3 (12 - 3abc) + 6abc
4³ = 58 - 3abc
and so
abc = (4³ - 58)/(-3) = -2
from which it follows that
a²b + a²c + ab² + b²c + ac² + bc² = 12 - 3 • (-2) = 18
Now we similarly simplify the extra terms in the 4th-degree expansion.
a³b + a³c + ab³ + b³c + ac³ + bc³
= a² (ab + ac) + b² (ab + bc) + c² (ac + bc)
= a² (ab + ac + bc) + b² (ab + ac + bc) + c² (ab + ac + bc) - a²bc - ab²c - abc²
= (a² + b² + c²) (ab + ac + bc) - (a²bc + ab²c + abc²)
= 10 • 3 - (a²bc + ab²c + abc²)
which means
4⁴ = a⁴ + b⁴ + c⁴
… … … … … … + 4 (30 - a²bc - ab²c - abc²)
… … … … … … + 6 (a²b² + a²c² + b²c²)
… … … … … … + 12 (a²bc + ab²c + abc²)
reduces to
136 = a⁴ + b⁴ + c⁴
… … … … … … + 6 (a²b² + a²c² + b²c²)
… … … … … … + 8 (a²bc + ab²c + abc²)
Now, we know a + b + c = 4 and abc = -2, so we have
a²bc + ab²c + abc² = abc (a + b + c) = -2 • 4 = -8
and so
136 = a⁴ + b⁴ + c⁴ + 6 (a²b² + a²c² + b²c²) + 8 • (-8)
200 = a⁴ + b⁴ + c⁴ + 6 (a²b² + a²c² + b²c²)
Finally, we know that ab + ac + bc = 3. Squaring this gives
(ab + ac + bc)² = 3²
a²b² + a²c² + b²c² + 2 (a²bc + ab²c + abc²) = 9
but we also know that a²bc + ab²c + abc² = -8, so
a²b² + a²c² + b²c² + 2 • (-8) = 9
a²b² + a²c² + b²c² = 25
Therefore, we end up with
200 = a⁴ + b⁴ + c⁴ + 6 • 25
⇒ [[[ a⁴ + b⁴ + c⁴ = 50 ]]
Select all expressions that show the distance between P and Q
DISCLAIMER THERE COULD BE MULTIPLE ANWSERS
A 5−(−8)
B 5+|−8|
C |−8+5|
D 5+(−8)
Answer:
A and B
Step-by-step explanation:
5-(-8) = 5+ 8= 13 (the distance between -8 and 5)
5+|-8| = 5 +8 = 13
Now simplify the expression.
(2x − 3)(5x + 6)
Answer:
10x+18
Step-by-step explanation:
2x*5x=10x
-3*6=18
10x+18=
x=1.8
Surya cut a pumpkin into three pieces. He sold two pieces that weighed about 1/2 and 1/3 respectively of the total weight of the pumpkin. If the remaining pumpkin weighed 600g. Which of these will be CLOSEST to the total weight of the pumpkin?
A. 1 KG
B. 3 KG
C. 3.6 KG
D. 4.8 KG
PLS ANSWER ASAP!!! ILL MAKE U BRAINLIEST♡
Answer:
b
Step-by-step explanation:
what are the first 100000000000000000000 digits of pie
Answer:
The first 10 digits of pi (π) are 3.1415926535. The first million digits of pi (π) are below, got a good memory? Then recite as many digits as you can in 30 ...
Step-by-step explanation:
ababccaaadadaddaddaa
Answer:
ababababababababbbababbbababbababaaaaaabababbababab
Step-by-step explanation:
hru and thank u for this question
Answer:
dsvrg
Step-by-step explanation:
Jeremy solved 35 problems, which is (5)/(7) of all his homework problems. How many problems are in his homework?
ASAP
Answer:
49
Step-by-step explanation:
make an equation:
5/7x=35
x=35*7/5
x=7*7
x=49
x is the total amount
Write a two-column proof.
I WILL GIVE BRAINLIEST TO BEST ANSWER! PLS HELP!
refer to images for question
MUST WRITE FULL PROOF
Proof:-
In ∆XYZ and ∆VWZ
[tex]\because\sf\begin{cases}\sf XZ\cong VZ(Given)\\ \sf YZ\cong WZ(Given)\\ \sf <XZY\cong <WZV(Opposite angles)\end{cases}[/tex]
Hence
∆XYZ[tex]\cong[/tex]∆VWZ(Side-Angle-Side)
Hey y’all can u please give me the correct answer.
Step-by-step explanation:
-8 - 6x < 1
Bringing like terms on one side
-6x < 1 + 8
-6x < 9
x < 9/-6
x < 3/-2
I think the correct answer is none
The price of a skateboard was reduced from $150 to $90. By what percentage was the price of the skateboard reduced?
Answer:
jjjjStep-by-step explanation:
The price tag on the remote-controlled helicopter said $79. If Jimmy buys the
helicopter, he'll also have to pay 8% sales tax. How much does the helicopter cost with
tar?
Step-by-step explanation:
79×0.08 = 6.32 (tax)
79(price) + 6.32(tax) = 85.32
Val has 3/4 gallon of milk. He gives 1/2 of it to a friend. How many gallons of milk does Val have left
Step-by-step explanation:
given, ¾-½= ¼gallon milk.
hope this helps you.
Ernie played 35 soccer games this season. He won 28 games. Enter the
percentage of games Ernie won.
Answer:
I think it;s 9.8
Step-by-step explanation:
You divide 35 and 28 i think
What is the relationship between 0.04 and 0.004?
Answer: 0.04 is exactly 10 times as much as 0.004. If we multiply 0.004 by 10, we will get 0.04 as an answer. In other words, we can say that 0.004 is exactly 1/10th of 0.04. It means that if we divide 0.04 by 10, we will get 0.004 as an answer.
Answer:
Look below
Step-by-step explanation:
0.04 means 4 hundredths since if you divide 4 by 100 you get 0.04. You can try it yourself on a calculator. 0.004 is 4 thousandths. Same thing with 0.04, if you divide 4 by 1000 you will get 0.004. Hope this helps! If you want me to explain a bit more I can try
Plz help me, i need this done 2 get my phone back
Answer:
Step-by-step explanation:
4x = 3yz^2
4x/3y = z^2
To undo an exponent you root it.
[tex]\sqrt{4x/3y} = z[/tex]
I believe this is it. you basically just have to divide by the numbers to get z by itself. If you have any questions please comment.
Grade 4c left their school at7:30 am for their trip to the market. They return at 10:45 am how much time did they spend onthe trip?
Answer:
3hrs 15 minutes
Step-by-step explanation:
10:45
-07:30
------------
03:15
------------
solve pls brainliest
Answer:
$30
Step-by-step explanation:
Over 5 years she will have been paid $30 in interest. To find the amount of interest first find the total amount the account has after 5 years. The formula for simple interest is A=P(1+rt). Where A is the total amount, P is the principal (initial amount), r is the rate in decimal form, and t is the time in years.
Plug in the values to solve, A=200(1+0.03*5). Then, simplify to get A=230. Finally, to get only the interest, I, use the formula A-P=I. So, 230-200=30. Therefore, the interest Leila earned was 30 dollars.
100 pounds of chocolate is packaged into boxes each containing 1^1/4 pounds of chocolate. Each box is then sold for $1.75. What is the total selling price for all of the boxes of chocolate?
Answer:
$140
Step-by-step explanation:
Total Amount of chocolate = 100 Pounds
Amount of chocolate in one box = 1 1/4 Pounds = 5/4 Pounds
Number of boxes = 100/(5/4) = 80 boxes
Cost of one box = $1.75
Total selling price = 80×1.75 = 140
The total selling price for all the boxes of chocolate is $140
Answer: $140
Step-by-step explanation:
Convert
1 1/4 --> 1.25
Solve
100/1.25= 80
80*1.75= $140
Consider the reaction. Upper n subscript 2 (g) plus 3 upper H subscript 2 (g) double-headed arrow 2 upper N upper H subscript 3 (g). At equilibrium, the concentrations of the different species are as follows. [NH3] = 0. 105 M [N2] = 1. 1 M [H2] = 1. 50 M What is the equilibrium constant for the reaction at this temperature? 0. 0030 0. 030 34 340.
The equilibrium constant of the reaction is 0.0030.
The equation of the reaction is given as;
N2(g) + 3H2(g) ⇆ 2NH3 (g)
We have the following information at equilibrium;
[NH3] = 0.105 M
[N2] = 1.1 M
[H2] = 1.50 M
Hence, we can calculate the equilibrium constant using the relation;
K = [NH3]^2/ [N2] [H2]^3
Substituting values;
K = [0.105]^2/[ 1.1] [1.50]^3
K =0.011025 /3.7125
K= 0.0030
Learn more: https://brainly.com/question/17960050
Describe the error in finding the measure of the angle.
According to the Rational Root Theorem, the following are potential roots of f(x) = 2x2 2x â€" 24. â€"4, â€"3, 2, 3, 4 Which are actual roots of f(x)?.
Rational root theorems are used to determine the potential roots of a polynomial function
The actual roots are 3 and -4The potential roots are [tex]\pm0.5, \pm1, \pm 1.5, \pm2, \pm 3, \pm 4, \pm 6, \pm 8, \pm12,\pm 24[/tex]The function is given as:
[tex]f(x) = 2x^2 + 2x - 24[/tex]
Expand
[tex]f(x) = 2x^2 + 8x - 6x - 24[/tex]
Expand
[tex]f(x) = 2x(x + 4) - 6(x + 4)[/tex]
Factor out x + 4
[tex]f(x) = (2x -6)(x + 4)[/tex]
Set to 0
[tex](2x -6)(x + 4) = 0[/tex]
Split
[tex](2x -6) = 0\ or\ (x + 4) = 0[/tex]
Make x the subject in both cases
[tex]2x = 6\ or\ x= -4[/tex]
Solve for x
[tex]x = 3\ or\ x= -4[/tex]
So, the actual roots are 3 and -4
Recall that, the function is:
[tex]f(x) = 2x^2 + 2x - 24[/tex]
For a function,
[tex]f(x) = px^n + ax^{n-1} +.......+q[/tex]
According to rational root theorem, the roots of f(x) are:
[tex]Roots = \pm \frac{Factors\ of\ q}{Factors\ of\ p}[/tex]
The roots of 24 are:
[tex]24 = \pm1, \pm2, \pm 3, \pm 4, \pm 6, \pm 8, \pm12,\pm 24[/tex]
The roots of 2 are:
[tex]2 = \pm1, \pm2[/tex]
So, the possible roots are:
[tex]Roots = \frac{\pm1, \pm2, \pm 3, \pm 4, \pm 6, \pm 8, \pm12,\pm 24}{\pm 1, \pm 2}[/tex]
This gives:
[tex]Roots = \pm1, \pm2, \pm 3, \pm 4, \pm 6, \pm 8, \pm12,\pm 24,\pm0.5, \pm1, \pm 1.5, \pm 2, \pm 3, \pm 4, \pm6\pm 12[/tex]
Remove repetitions
[tex]Roots = \pm0.5, \pm1, \pm 1.5, \pm2, \pm 3, \pm 4, \pm 6, \pm 8, \pm12,\pm 24[/tex]
Read more about rational root theorem at:
https://brainly.com/question/11015301
3 and -4 dudes and dudettes
edg2022
Determine whether the data in the table can be modeled by a linear equation. Explain.
If possible, write a linear equation that represents y as a function of x
Answer:
yes, because they're varying in a constante rate
y = -1/2x +3.5
how many solutions does x squared= 4x -5 have
Answer: There are no solutions
Answer:
Two
Step-by-step explanation:
x^2 = 4x - 5 is equivalent to x^2 - 4x + 5 = 0 in standard quadratic form.
All quadratics have exactly two roots/solutions.
In this case the coefficients of the polynomial are 1, -4 and 5, and so the discriminant is b^2 - 4ac, or (-4)^2 - 4(1)(5), or 16 - 20 = -4.
Since the discriminant is negative, we conclude that this polynomial has two complex roots/solutions.
find the common factor and rewrite.
8A + 11A =
i've been on this question for like 3 days. the answer is NOT 1(8A+11A)
Answer:
8A + 11A = A(19)
Step-by-step explanation:
8A + 11A
= A(8 + 11) Factor out the common factor A
= A(19)
. A circulating pump consumes 450 W for 24 hr a day. a. Determine the energy consumed for a one-year period. b. Calculate the utility energy charges for this period at a rate of $0.12/kWh. c. Determine the energy saved for a one-year period, if the pump is switched off 8 hr a day. d. Calculate the energy cost savings for this period at a rate of $0.12/kWh, if the pump is switched off 8 hr a day.
Answer:
a. 3942 kWh
b. $473.04
c. 1314 kWh
d. $157.68
Step-by-step explanation:
a.There are 1000 W in 1 kW, so 450 W = 0.450 kW. The energy used per day is ...
(0.45 kW)(24 h) = 10.8 kWh . . . . energy per day
Then in a 365-day year, the energy used is
(365 da/yr)(10.8 kWh/da) = 3942 kWh/yr
__
b.At the rate of $0.12/kWh, the cost of running the pump is ...
($0.12/kWh)(3942 kWh/yr) = $473.04/yr
__
c.Switching the pump off for 1/3 of the time will save 1/3 of the energy found in part (a):
1/3(3942 kWh) = 1314 kWh . . . . energy saved by switching off the pump
__
d.The savings will be 1/3 of the cost of running the pump full time:
1/3($473.04/yr) = $157.68/yr
THE WHOLE THING ALL OF IT PLEASE I WILL MAKE U BRAINLIEST!!
Answer:
3x+8=17 2x+8=2(x+4) x+4=x-4
3x=17-8 2x+8=2x+8 No solution
3x/3=9/3 All Real numbers
x= 3
infinitely many solution
Q2) It means this equation has no answer to it and no solution
Q3) x+ 4=x-4 has no solution
Q4) It means that it has many answers to it and many steps to it to get the answer
Q5) By doing the answer for example 3x+9=17 it has many solutions and we can get the answer by
I need the answers to a b c and d PLZ DUE AT 11:59PM ITS 9:23PM
Answer:
do u even still need the answer its been aweek since you asked it and it was due that same day
Step-by-step explanation:
What is the interest on $600 when it is placed for four years in a bank account that gives 2% simple interest
Answer:
12, or if its yearly, 48.
I NEED HELP I NEED HELP I NEED HELP I NEED HELP I NEED HELP
What do you mean? What’s the question?
Answer:
Unknown
Step-by-step explanation:
Question, not provided.
is the constant rate increasing or deceasing
The leader of the Protestant Reformation was a German monk called ___.
