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 ]]
Find the slope for (4,5)and (2,3)
Answer:
1
Step-by-step explanation:
I need help with this question,
Which point is located in Quadrant III ?
(−1/2, 1/4)
(1/2, −1/4)
(−1/2, −1/4)
I will give you brainliest if you are right
Answer:
here you go :)
Quadrant lll is always both negative so the answer is (-1/2, -1/4)
Step-by-step explanation:
Jesse collected 42 football cards and 28 baseball cards. Simplify the ratio of baseball cards to football cards in his collection
Answer:
3:2
Step-by-step explanation:
42:28
21:14
3:2
Help please Pretty plz
Answer:1/10
Step-by-step explanation:
Start by multiplying 1/100 by 1/10
The answer will be 1/1000
A student says that it 5x^2 =20,then x must be
equal to 2. Do you agree or disagree with the
student? Justity your answer.
Hope you could get an idea from it.
Doubt clarification - use comment section.
The slope of a line is (-1/2) and a point on the line is (-2, 3). Find the equation for the line in slope intercept form.
Answer:
y=-1/2x+2
Step-by-step explanation:
y-y1=m(x-x1)
y-3=-1/2(x-(-2))
y-3=-1/2(x+2)
y=-1/2x-2/2+3
y=-1/2x-1+3
y=-1/2x+2
Hey y’all can u give me the correct answer pls.
Answer:
x > -7 (Hey man think before ask)
Step-by-step explanation:
It is so easy .....
Answer:
-6,-4,-2,3
Equation: x>−7
Step-by-step explanation:
Just ask.
Hope this helps.
make q the subject of the formula
Answer:
q = 2p - 1/3
Step-by-step explanation:
2q + 2p = 1+5q
2q = 1+5q-2p
-3q=1-2p
-q = 1-2p/3
q= 2p-1/3
Answer:
q=[tex]\frac{2p-1}{3}[/tex]
Step-by-step explanation:
2(q+p)=1+5q
EXPAND BRACKET
2q+2p=1+5q
MINUS 2q FROM BOTH SIDES
2p=1+3q
MINUS 1 FROM BOTH SIDES
2p-1=3q
DIVIDE BOTH SIDES BY 3
[tex]\frac{2p-1}{3}[/tex]=q
2x+x+x+2y times 3y times y
Answer:
4x+ 6y^2
Step-by-step explanation:
Add 3 5/6+(−1 1/6). Write your answer as a mixed number in the simplest form.
The result of 3 5/6+(−1 1/6) is 2 2/3.
INTRODUCTIONThe definition of a fraction is that the division of two integers by the number being divided is called the numerator and the divisor is called the denominator.
Kinds of Fractions:
a. Mixed fractions, for example: 1⅘, 2½, 3⅔, etc.b. Common fractions, for example: , , , etc.c. Percent fractions, for example: 50%, 2%, 6%, etc.d. Decimal fraction, for example: 0.50DISCUSSION3 5/6+(−1 1/6)
= 23/6 + (-7/6)
= 23/6 - 7/6
= 16/6
= 8/3
= 2 2/3
Conclusion:So, The result of 3 5/6+(−1 1/6) is 2 2/3.
Learn more:mampir ke indo :v
Example of Fraction Problem: https://brainly.co.id/task/14280048Example of Fraction Subtraction Problem: https://brainly.co.id/task/23345402Material about fractional equations: https://brainly.co.id/task/3304077______
Answer Details:Grade: 4 Elementary SchoolSubject: MathMaterial: Chapter 6 -ShardsCategorization Code: 4.2.6Keywords: Subtract mixed fractions.
[tex]2\frac{2}{3}[/tex] is the simplest form of expression [tex]3\frac{5}{6} +(-1\frac{1}{6} )[/tex]
What is Fraction?A fraction represents a part of a whole.
Fractions are three types proper fraction, improper fraction and Mixed fraction.
The given expression is [tex]3\frac{5}{6} +(-1\frac{1}{6} )[/tex]
Three five by six plus minus one one by six.
Let us convert mixed fraction to improper fraction and solve
[tex]3\frac{5}{6} +(-1\frac{1}{6} )[/tex]
(18+5)/6 -(6+1)/6
23/6-7/6
Now take LCM as 6
(23-7)/6
Sixteen by six
16/6
Eight by three
8/3
[tex]2\frac{2}{3}[/tex]
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What is the value of q in the equation 49 - 2 = 39+8?
q
=
10
0 -
7
10
7
06
0 10
Answer:
The is no "q" in the equation provided. If there is supposed to be, please add it. Otherwise, q is undefined. It can have any value. I choose 0.
Step-by-step explanation:
help me with this!!!
3 tickets to the museum cost $12.75. At this rate, what is the cost of:
1 ticket?
5 tickets?
Answer:
1 ticket would be $4.25
5 tickets would be $21.25
Step-by-step explanation:
Hope that help, mind if a get a brainliest now?
Answer:
4.25 for 1 ticket, 21.25 for 5
Step-by-step explanation:
12.75 divided by 3 = 4.25 x 5 = 21.25
when a number is decreased by 3% the result is 80. what was the original number?
Answer:
82.47
Step-by-step explanation:
.93n = 80
n = 80/.93
n = 82.47
[tex]\huge \dag \sf{Answer \: it}[/tex]
[tex]solve \: for \: x \\ \\ {2}^{ {x}^{2} } 2 \frac{3x \sqrt{3} }{2} = 8 [/tex]
Don't spam and Don't copy.
Need answer with explanation
Don't answer if you don't know
Best of luck guys
Answer:
Answer: x = √3 and 8/(3√3)
Step-by-step explanation:
[tex]{ \rm{({2}^{ {x}^{2} } )(2 \frac{3x \sqrt{3} }{2}) = 8}}[/tex]
• For the first bracket:
[tex]{ \rm{ {2}^{ {x}^{2} } = 8}} \\ \\ { \rm{ {2}^{ {x}^{2} } = {2}^{3} }} \\ \\ { \rm{ {x}^{2} = 3}} \\ \\ { \rm{x = \pm \sqrt{3} }}[/tex]
• For the second bracket;
[tex]{ \rm{ 2 \frac{3x \sqrt{3} }{2} = 8}} \\ \\ { \rm{3x \sqrt{3} = 8 }} \\ \\ { \rm{3x = \frac{8}{ \sqrt{3} } }} \\ \\ { \rm{x = \frac{8}{3 \sqrt{3} } }}[/tex]
what is 2+2 i need help like rly bad i will give 100 points
Answer:
4 bestie <3
Step-by-step explanation:
Answer:
Step-by-step explanation:
4
Isolate f variable! Solve the equation algebraically! Show work!!
42 = 6f
Answer:
f=7
Step-by-step explanation:
42=6f
/6 /6
42/6=7
You divide 6 from 42, which leaves you with 7=f.
Pls help me
4(3-x)=6(x+2)+20
Answer:
x = -2
Step-by-step explanation:
11,046 as a decimal
Answer:
It cant be converted to decimal or fraction while it is a integer
Define algebra ? And give some examples..
Answer:
Algebra is a abstract way of thinking about things. We usually see this by defining things we don't know as some sort of symbol (Matter of fact. All of Mathmatics revolves around symbols). It doesn't necessarily have to be a number. I could label some big brown sturdy thing with green stuff on top as a tree which in a mathmatical sense the tree would be the variable and the actual thing would be the thing inside of the variable. But basically it's an abstract way of thinking
Step-by-step explanation:
What number multiplied by 3/5 has a product of 1?
Answer:
5/3
Step-by-step explanation:
let the number be x
3/5×[x] = 3x/5
3x/5=1
x=1÷3/5
x=5/3
-10 + = -3 I need help
Answer:
the answer is
7
Step-by-step explanation:
why is it 7 it is cause -10+7=3
Answer:
7
the answer is 7 because -10+7=-3
A plant grows at a constant rate. The table shows the heights of the plant for different numbers of days of growth. Which of the following pairs of days and heights could also be included in the table? Select all that apply.
What is the equation of the line that passes
through the points (3,-1) and (7, 3)?
Answer: y = 1x + -4
Equation of a straight line:
y = mx + b ------(i)
Step by Step Solution:
Step 1: Calculating Slope (m).
m = y2-y1 /x2-x1
m = 3--1 /7-3
m = 4/4
m = 1
Now putting value of m in equation (i)
y = 1x + b -----(ii)
Step 2: Calculating Y-intercept (b).
Lets choose the first point, (3,-1) for calculating y-intercept:
y = mx + b
-1 = 1(3) + b
-1 = 3 + b
-4 = b
b = -4
Now putting value of b in equation (ii)
y = 1x + -4
How much does Janet earn per hour
Plz helppppp
Answer:
$7.50 per hour
Step-by-step explanation:
after 2 hours, she earns $15
after 4 hours, she earns $30
after 8 hours, she ears $60
15/2 = 30/4 = 60/8 = 7.5/1
Given that the shape is a kite and Angle A=25, Angle C=85, what is Angle
D?
Answer:
m∠D = 165°
Step-by-step explanation:
∠B and ∠C are congruent, so ∠B = 85°
The angles of the kite all added together have to add up to 360°
85 + 85 + 25 + x = 360
195 + x = 360
x = 165
Answer:
m∠D = 165°
Step-by-step explanation:
∠B and ∠C are congruent, so ∠B = 85°
The angles of the kite all added together have to add up to 360°
85 + 85 + 25 + x = 360
195 + x = 360
x = 165
pls help i need help
[tex](5^6)^0 \\\\=5^0\\\\=1[/tex]
can somebody help me out which one is correct
A , B , C , D ?
PLEASE HELP!! name the property of equality that the statement illustrates:
if x = y , then -2x = -2y
The multiplication property of equality states that if [tex]x=y[/tex] is true, then [tex]xz=yz[/tex] must also be true.
The property of equality illustrated by the statement "if x = y, then -2x = -2y" is the: Multiplication Property of Equality.
How to Identify the Property of Equality?The properties of equality describe the relation between the two sides of an equation and state that the two sides remain equal even after applying the same arithmetic operation on each side.
The property of equality illustrated by the statement "if x = y, then -2x = -2y" is the Multiplication Property of Equality.
This property states that if you multiply both sides of an equation by the same nonzero number, the equality still holds true.
In this question, both sides of the equation are multiplied by -2, maintaining the equality between the expressions.
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