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 ]]
jerimiah can type 300 words in 1/5 of an hour. How many words can he type in 1 hour
Answer:
1500 words
Step-by-step explanation:
300 times 5 equals 1500.
Divide. Round your answer to the nearest thousandth.
2,891.3030 - 0.728=
Answer:
3971.570
Step-by-step explanation:
Which of the following is equal to 42.736 ? Select all that apply
Answer:
sorry but i need more information on the question to answer it.
Step-by-step explanation:
Hope this helps you!
can i please have help with this question
brainliest for answer
Answer:
yes I can help
Step-by-step explanation:
which one please
I need help with this question,
A boy is walking at 3 m/s He is 25 kg and his back pack is 5kg. What is the boys momentum when carrying the back pack?
90 kg m/s
10 kg m/s
15 kg m/s
75 kg m/s
Answer:
90 kg m/s
Step-by-step explanation:
The equation for momentum is
p= mass x velocity
so our velocity is 3 thats is given
the mass must be calculated by adding 25 and 5
this adds to 30
30 x 3 = 90
what is the exact circumference of the circle? A. 10 FT B. 20 FT C. 40 FT D. 60 FT
Answer:
D. 60 Ft
Step-by-step explanation:
circumference = πd
π 20
60ft
Answer:
Hopefully others can use mine and get it correct!
Step-by-step explanation:
I just took this and got it wrong by listening to the other and went with 60 FT nope its 20FT
Foundations in Algebra: Probability Assignment
1. Andrew claims:
A spinner has 4 sections -
red, yellow, green, and blue.
The probability of getting the
red section is 0.25.
Is he correct? Explain your answer fully:
I
Answer:
yes a 1/4 probability is 0.25
Describe the correlation of the following data:
strong, positive
strong, negative
weak, positive
weak, negative
Answer:
Weak and positive
Step-by-step explanation:
I just said it
The correlation of the data provided is Weak and the positive option (C) is correct.
What is a regression line?A regression line is a graph that illustrates the pattern of a group of statistics. In other words, it shows the data's best pattern.
We have a scatter plot shown in the picture.
As we can see in the dot plot the dots are scattered so much so there will be a very weak correlation between the variables.
As we know, the correlation is the relation between two variables which is a quantitative type and gives an idea about the direction of these two variables.
The formula for the regression coefficient:
[tex]\rm r = \dfrac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{{[n\sum x^2- (\sum x)^2]}}\sqrt{[n\sum y^2- (\sum y)^2]}}[/tex]
Drawing a line of best fit, we will see the correlation is Weak and positive.
Thus, the correlation of the data provided is Weak and the positive option (C) is correct.
Learn more about the regression line here:
brainly.com/question/7656407
#SPJ2
Hey y’all , please give me the correct answer.
Answer:
-s-6>-7
Step-by-step explanation:
Just ask!
Hope this helps. :)
Answer:
-s-6>-7 (top right)
Step-by-step explanation:
s = -18 > means what is on the left is bigger than what is on the right
top right one = - s - 6 > - 7
top right one = - ( - 18) - 6 > -7
top right one = 18 - 6 > -7
top right one = 12 is bigger than -7
Someone please answer this quickly!
Note: T is the center of the circle.
Here is my answer. I hope it is helpful.
Write the following statements in the form of equations.
If you subtract 6 from 3 times a number, you get 18.
Will mark brainliest for this question
Answer:
a
Step-by-step explanation:
It's the positive answer
3 is what percent of 50
Answer:
6%
Step-by-step explanation:
3/50 x 100/1% = 3/2 x4/1% = 12/2%=6%
WILL GIVE BRAINLIEST
Step-by-step explanation:
The slope-intercept form of an equation of a line:
[tex]y=mx+b[/tex]
[tex]m[/tex] - slope
[tex]b[/tex] - y-intercept
[tex]11)\ m=-1,\ b=2\\\\y=-x+2\\\\12)\ m=-\dfrac{3}{2},\ b=3\\\\y=-\dfrac{3}{2}x+3\\\\13)\ m=3,\ y=-2\\\\y=3x-2\\\\14)\ y=\dfrac{3}{4},\ b=1\\\\y=\dfrac{3}{4}x+1\\\\15)\ m=\dfrac{1}{2},\ b=1\\\\y=\dfrac{1}{2}x+1\\\\16)\ m=-\dfrac{2}{5},\ b=0\\\\y=-\dfrac{2}{5}x\\\\17)\ m=-7,\ b=2\\\\y=-7x+2\\\\18)\ m=\dfrac{4}{3},\ b=-4\\\\y=\dfrac{4}{3}x-4[/tex]
19) and 20) you have in other your question.
AHHHHHH HELP PLEASE
[tex]3.25 \div \sqrt{24} [/tex]
Find the quotient of 1/6 and 2/9
Answer:
3/4 or 0.75
Step-by-step explanation:
When dividing- quotient- fractions, remember keep, change, flip
Keep- keep the first fraction
Change- change the division sign into a multiplication sign
Flip- flip the second fraction
So your problem really is:
[tex]\frac{1}{6} *\frac{9}{2}[/tex]
Which is: 9/12 or 3/4
Given that T is the centroid of △DEF and FT=12
The centroid of a triangle divides the median of the triangle into 1 : 2
The measure of FQ is 18, while the measure of TQ is 6
Because point T is the centroid, then we have the following ratio
[tex]\mathbf{TQ : FT =1 : 2}[/tex]
Where FT = 12.
Substitute 12 for FT in the above ratio
[tex]\mathbf{TQ : 12 =1 : 2}[/tex]
Express as fraction
[tex]\mathbf{\frac{TQ }{ 12} =\frac{1 }{ 2}}[/tex]
Multiply both sides by 12
[tex]\mathbf{TQ =\frac{1 }{ 2} \times 12}[/tex]
This gives
[tex]\mathbf{TQ =\frac{1 2}{ 2}}[/tex]
Divide 12 by 2
[tex]\mathbf{TQ =6}[/tex]
The measure of FQ is calculated using:
[tex]\mathbf{FQ = FT + TQ}[/tex]
Substitute 12 for FT, and 6 for TQ
[tex]\mathbf{FQ = 12 + 6}[/tex]
Add 12 and 6
[tex]\mathbf{FQ = 18}[/tex]
Hence, the measure of FQ is 18, while the measure of TQ is 6
Read more about centroids at:
https://brainly.com/question/11891965
Find the value of x so that the function has the given value.
n(x) = 5x + 8; n(x) = 23
The range of f(x) = |x| is y ≥ 0. If a < 0 and b ≠ 0 for g(x) = a|x| + b, what is the range of function g?
Answer:
blue or maybe green try purple
the two polygons are similar. Write a proportion and solve for x
SHOW WORK PLEASE
Answer:
Answer: 2.5
Step-by-step explanation:
It seems as if everything is divided by two so 5/2 is 2.5
Use each model to predict the life expectancy of residents of a country for which the average annual income is $80,000
The life expectancies of residents of a country for which the average annual income is $80,000 for the three models are 12309.9352, 172.2436 and 4828.1393
The life expectancies of the models are given as:
[tex]y =69.9352 + 0.1530\times (income)[/tex] --- model 1
[tex]y =4.2436 + 0.0021\times (income)[/tex] --- model 2
[tex]y =4.1393 + 0.0603\times (income)[/tex] --- model 3
Given that the average annual income is $80,000;
We simply substitute 80000 for income in the equations of the three models.
So, we have:
Model 1
[tex]y =69.9352 + 0.1530\times (income)[/tex]
[tex]y =69.9352 + 0.1530\times 80000[/tex]
[tex]y =12309.9352[/tex]
Model 2
[tex]y =4.2436 + 0.0021\times (income)[/tex]
[tex]y =4.2436 + 0.0021\times 80000[/tex]
[tex]y =172.2436[/tex]
Model 3
[tex]y =4.1393 + 0.0603\times (income)[/tex]
[tex]y =4.1393 + 0.0603\times 80000[/tex]
[tex]y =4828.1393[/tex]
Hence, the life expectancies are 12309.9352, 172.2436 and 4828.1393
Read more about linear models at:
https://brainly.com/question/8609070
Type the correct answer in the box.
3 units
2 units
2 units
The area of the figure is
square units.
Reset
Next
Plz help
Answer:
8 units
Step-by-step explanation:
Area of square = 2x3 = 6 units
Area of triangle = (2x2)/2 = 2 units
Total area = 6 +2 = 8 units
Answer:
As previous answer stated, 8 units
Step-by-step explanation:
You multiply one side of the square by another to get 6 units, then, add the units from the triangle
please help me im only looking for real answers
if you put a link to an inappropriate site i WILL report it
Answer: y=3/2x
Step-by-step explanation:
First, to find the equation, we can figure out that the cost depends on how many tickets are bought. We can use cost as our y and # of tickets as our x.
An equation in slope intercept form is as follows:
y=mx+b
To get our slope, we must figure out the relationship between the cost and the number of tickets, or the y and the x.
We can use the slope formula, [tex]\frac{y2-y1}{x2-x1}[/tex], to find the slope, m.
[tex]\frac{3-6}{2-4}[/tex] = [tex]\frac{3}{2}[/tex]
Thus, the slope of the equation is 3/2, making our final equation be y=3/2x.
Darren got 42 out of 56 correct on a test. What percent of the questions did he get correct?
Answer: Darren got 75% of the questions correct.
Step-by-step explanation:
To find the percentage of questions Darren got correct, we can divide the questions he got correct by the total number of questions.
42 ÷ 56 = 0.75
0.75 X 100 = 75
So, Darren got 75% of the questions correct.
hello guys .. can y’all help me with this question and explain it to me ? please and thank you:) 25 POINTS !
Answer:
B, 5.4
Step-by-step explanation:
BC = 36
FE = 7.2
AB = 27
x = Unknown value
They are proportional so...
[tex]\frac{36}{27} =\frac{7.2}{x}[/tex]
x = 5.4
Find the difference.
5 and two-thirds minus 2 and one-half equals what number?
A. three and one-sixth
B. three and two-thirds
C. three and one-half
D. three and five-sixths
Answer:
A
Step-by-step explanation:
5 2/3 - 2 1/2 = 3 2/3 - 1/2 = 3 4/6 - 3/6 = 3 1/6
If x^2 - 6x +8=0, then x-4=0 or x-2=0.
Step-by-step explanation:
X - 4 = 0
X = 4
X - 2 = 0
X = 2
So the values of x are 2 and 4
al dividir A entre 23 se obtuvo 247 de cociente y el residuo fue el maximo. hallar A a)5242 b)5436 c)5703 d)5332 solo esas 4 alternativas tienen que salirte y respondan con resolucion plss
Aplicando la división, hay que el valor de A es dado por:
c) 5703
Una división es representada por:
[tex]\frac{a}{b} = c + \frac{r}{b}[/tex]
En que:
c es el cociente.r es el residuo.En este problema, [tex]b = 23[/tex], y también:
Cociente de 247, o sea, [tex]c = 247[/tex].Residuo maximo, que es 22, o sea, [tex]r = 22[/tex].[tex]\frac{a}{b} = c + \frac{r}{b}[/tex]
[tex]\frac{A}{23} = 247 + \frac{22}{23}[/tex]
[tex]A = 23(247) + 22 = 5703[/tex]
El valor de A es dado por:
c) 5703
Un problema similar, que tambíen envuelve división, es dado en https://brainly.com/question/14270856
Please help serious answers only
Answer:
#4 x = 3 ± i√31
2
Step-by-step explanation:
Solve by completing the square
x² - 3x + 10 = 0
x² - 3x = -10
x² - 3x + 9/4 = -10 + 9/4 half of 3 and square the answer (half of 3 = 3/2
and square the number (3/2)² = 9/4
x² - 3x + 9/4 = -31/4
[tex](x - \frac{3}{2} )^2 = \frac{-31}{4}[/tex]
[tex]\sqrt{(x-\frac{3}{2})^2 } =\sqrt{\frac{-31}{4} }[/tex]
[tex]x - \frac{3}{2} = \frac{i\sqrt{31} }{2}[/tex]
x = 3/2 ± i[tex]\sqrt{31}[/tex] /2
x = 3 ± i√31
2