Let
[tex]S = \displaystyle \sum_{1\le j<k\le673} z_jz_k[/tex]
and
[tex]p_{\mu,\nu}(x) = (x-z_1)^\mu (x-z_2)^\mu \cdots (x-z_\nu)^\mu[/tex]
The [tex]z_jz_k[/tex] terms occur in the coefficient of the [tex]x^{\mu\nu-2}[/tex] term, given by
[tex]\displaystyle \left[x^{\mu\nu-2}\right] p_{\mu,\nu} = \binom\mu2 \sum_{r=j}^\nu z_j^2 + \mu^2 \sum_{1 \le j < k \le \nu} z_jz_k[/tex]
(essentially due to Vieta's formulas)
With [tex]\mu=3[/tex] and [tex]\nu=673[/tex], we have
[tex]\displaystyle \left[x^{2017}\right] p_{3,673} = \binom32 \sum_{j=1}^{673} z_j^2 + 3^2 S \\\\ ~~~~ \implies S = \frac{19}9 - \frac13 \sum_{j=1}^{673} z_j^2 \\\\ ~~~~ \implies S = \frac{19}9 - \frac13 \left(\left(\sum_{j=1}^{673} z_j\right)^2 - 2 S\right) \\\\ ~~~~ \implies S = \frac{19}3 - \left(\sum_{j=1}^{673} z_j\right)^2[/tex]
The remaining sum on the right is the sum of the roots of [tex]\sqrt[3]{p_{3,673}(x)} = p_{1,673}(x)[/tex].
Recall that the sum of the roots of a polynomial
[tex]P_n(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0[/tex]
is [tex]-\frac{a_{n-1}}{a_n}[/tex]. Then the sum of the roots of [tex]p_{3,673}(x)[/tex] is -20, which is 3 times the sum of the roots (counting multiplicity) of [tex]p_{1,673}[/tex], so
[tex]\displaystyle \sum_{j=1}^{673} z_j = -\dfrac{20}3 \\\\ ~~~~ \implies S = \frac{19}3 - \left(-\frac{20}3\right)^2 = -\frac{343}9 \\\\ ~~~~ \implies |S| = \frac{343}9, m=343, n=9 \implies m+n = \boxed{352}[/tex]
Find the Distance between the two points . (2,5) and (-4,7).
Answer:
6.32
Step-by-step explanation:
They gave you everything you need in the question - just plug the x and y values in where they showed you. (2, 5) is your (x1, y1) and (-4, 7) is your (x2, y2)
Plot the inequality on the number line.
x≤1 or x>8
The attached graph represents the graph of the inequality x ≤ 1 or x > 8
What are inequality expressions?Inequality expressions are mathematical statements that are represented by variables, coefficients and operators where the opposite sides are not equal
How to plot the inequality on the number line?The inequality expression is given as
x ≤ 1 or x > 8
When an inequality is represented with < or >, then we make use of the open circle to represent the inequality.
Otherwise, we make use of the closed circle to represent the inequality.
Using the above as a guide, we have the following representations:
x ≤ 1 -> Closed circle
x > 8 -> Open circle
See attachment for the graph of the inequality
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find the measure of the angle indicated x + 96 x+ 96
The measure of the angles shown is 90 degrees
Lines and anglesA line is defined as the shortest distance between two points while the point where two lines meet is known as an angle.
From the given diagram, the sum of both adjacent angles is supplementary to have:
x + 96 + 96 + x = 180
2x + 192 = 180
2x = 180 - 192
2x = -12
x = -6
Substitute the value of x into the expression
<d = x + 96
<d = -6 +96
<d = 90 degrees
Hence the measure of the angles shown is 90 degrees
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Find an equation of the line that satisfies the given conditions. Through (1, −6); parallel to the line x + 2y = 6
y=1/2x-13/2 is equation of the line that satisfies the given conditions. Through (1, −6) and parallel to the line x + 2y = 6
What is slope intercept of line?y=mx+c
where m is the slope.
Firstly we need to find the slope of the line x - 2y = 6 by using slope intercept form, y=mx+c.
-2y=6-x
2y=x-6
y=1/2 x-6
m=1/2 slope of the line x - 2y = 6.
For parallel lines slope will be same.
Now Use y = mx + b and the point (1,-6) to find b, the y-intercept.
-6= (1/2)*1 + b
-6-1/2=b
-13/2=b
y=1/2x-13/2 is the equation of line.
Therefore y=1/2x-13/2 is the line parallel to x + 2y = 6.
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If y= -252 when x = 84, what is Y when X =74?
Answer:
Step-by-step explanation:
So lets turn this into a ratio
-252/84
so now we need to know
y/74
So, -252/84= -3
so y/74 needs to be equal to -3
74*-3 = 222
-222/74=-3
So our answer is y = -222
Hope this helps!!^^
Use the GCF and the Distributive Property to find the sum of 38 + 57
Answer:
95
Step-by-step explanation:
You want the sum of 38 and 57 by making use of the GCF and the distributive property.
GCFThe GCF of two numbers can be found by looking at their difference (or their remainder when one divides the other). Here, that is ...
57 -38 = 19
This value is a factor of both numbers, so is their GCF.
Distributive propertyThe distributive property tells us how to express the sum with the GCF factored out:
38 +57 = 19(2 +3)
Then the sum is ...
19(2 +3) = 19(5) = 95
The sum is 95.
How many quarts of pure antifreeze must be added to 3 quarts of a 10% antifreeze solution to obtain a 30% antifreeze solution?
_quart(s) of pure antifreeze must be added.
There are 0.857 quarts of pure antifreeze that must be added to 3 quarts of a 10% antifreeze solution to obtain a 30% antifreeze solution.
What is the percentage?
The percentage is defined as a ratio expressed as a fraction of 100.
We have been given that there is 10% times 3 quarts or 0.3 quarts of antifreeze in the current mix and we want a 30% solution.
Let the amount of pure antifreeze to be added would be x
The equation to solve becomes,
⇒ (0.3 + x )/(3 + x) = 0.3
⇒ (0.3 + x ) = 0.3 × (3 + x)
⇒ (0.3 + x ) = 0.9 +0.3x
⇒ 0.7x = 0.6
⇒ x = 0.857
Therefore, we need to add 0.857 quarts of pure antifreeze to the current solution to make a 30% solution.
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Pls help!!
Write and solve the problem. Type your answer into the answer box. -2 2/3 + 1 4/5
Answer: If i did it right it should be -0.866
Step-by-step explanation:
Which point is an x-intercept of the quadratic function f(x) = (x + 6)(x – 3)?
(0,6)
(0,–6)
(6,0)
(–6,0)
Answer:
( -6 , 0 )
Step-by-step explanation:
the other one would be ( 3 , 0 ) but hope this helps !!
U and V are mutually exclusive events.
P(U) = 0.29
P(V) = 0.38
Part (a)
P(U AND V) =
Part (b)
P(UV)=
Part (c)
P(U OR V)=
Answers
P(U and V) = 0
P(U or V) = 0.67
========================================
Explanation:
P(U and V) = 0 since U and V are mutually exclusive. Such events cannot happen at the same time. Example: flipping a coin to get heads and tails at the same time.
-------
P(U or V) = P(U) + P(V) - P(U and V)
P(U or V) = 0.29 + 0.38 - 0
P(U or V) = 0.67
-------
I'm not sure about P(UV) since there doesn't seem to be enough information. Does your teacher mention if the events are independent or not?
K
2
Create a proportion to solve for x.
16
22
11
A pease help please mb
solve for v -4v=-24 simplify your answer as much as possible
this is Multiplicative property of equality with integers
Answer:
v = 6
Step-by-step explanation:
Given equation:
-4v = - 24Divide both sides by - 4:
-4v / - 4 = - 24 / -4v = 6Answer:
The value of v is 6.
Step-by-step explanation:
Solving for the value of v,
→ -4v = -24
→ v = (-24)/(-4)
→ v = 24/4
→ [ v = 6 ]
Hence, the value of v is 6.
uijuguiobhj,mn,mgvplsssss
Step-by-step explanation:
adding 93 to itself 6 times
dividing 558 by 6
Answer:
3
Step-by-step explanation:
graph a function with a domain of all real numbers less than -4?
y
The range is the set of y values.
PLEASE HELP !
You go to a store to buy sneakers. You have a coupon for $15 off. Write a function f(d) that models this discount on an item that costs d dollars
The function f(d) that models this discount on an item that costs d dollars is f(d) = d - 15
How to write a function f(d) that models this discount on an item that costs d dollars?The given parameters are
Discount = $15
Value = d
The function f(d) that models this discount on an item that costs d dollars is
f(d) = Value - Discount
This gives
f(d) = d - 15
Hence, the function f(d) that models this discount on an item that costs d dollars is f(d) = d - 15
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Please explain what are integers?
Answer:
: integers
1.
a number which is not a fraction; a whole number.
a company sold yogurt in 120 g pots and one-eight of the content of the pots was chopped fruit, it is then decided to make the pots of yogurt 2/3 of their original weight with 1/5 fruit. will the company use more or less fruit per new design?
Answer:
More fruit for the new design.
Step-by-step explanation:
120 times 2/3 is 80.
1/5 of 80 is 16.
1/8 of 120 is 15.
15 < 16, so the second design has more fruit.
how to sketch a plot of the line x=−1 .
Answer:
See explanation
Step-by-step explanation:
Find the Domin and range f(x) =-2(x+3)2 -5
Domain and range of [f(x) =-2(x+3)2 -5] are as follows: x in RR, y in [-∞, -5].
What in mathematics are domain and range?The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f are contained in this collection (x). A function's range is the range of values that it can accept as input. The function produces this list of values once we enter an x value.
according to given information;
This is a polynomial of degree 2, "domain is x in RR," and "to determine the range, we require the vertex and if maximum" or "minimum turning point," the equation of the polynomial's vertex and range."" a parabola in "color(blue)"vertex form" is.
•(x)y=a(x-h)²+k
"where "(h,k)" are the coordinates of the vertex and a"
"is a multiplier"
f(x)=-2(x+3)²-5" is in this form"
"with "(h,k)=(-3,-5)" and "a=-2
"since " a <0" then maximum turning point"
"range is "y in(-oo,-5]
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Help please I don’t get it
Answer:
Step-by-step explanation:
-5 - 9x = 3-9x-8 (Used distributive property)
-5 - 9x = -9x-5 (-8+3 = -5)
- 9x = -9x (Add 5 to both sides)
0 = 0 (Add 9x to both sides)
As an introduction to probability, a student is asked to roll
a fair, six-sided number cube seven times. The results of
those seven rolls are shown below.
1, 4, 4, 4, 4, 6, 5
What is the range of the data?
01
1.53
4
5
Answer:
Step-by-step explanation:
Based on the calculations, the IQR of this data is equal to 2.
How to determine the IQR of this data?
IQR is an acronym for interquartile range and it is a measure of the middle 50% of data values when they are ordered from lowest to highest.
Next, we would order the data from lowest to highest and then determine the median as follows:
Data = {1, 4, 4,} 4, {4, 6, 5}
Median = 4.
For Q₁ of the lower half of the data, we have:
Lower half = {1, 4, 4,}
Q₁ = 4.
For Q₃ of the upper half of the data, we have:
Upper half = {4, 6, 5}
Q₃ = 6.
Mathematically, IQR is the difference between quartile 1 (Q₁) and quartile 3 (Q₃):
IQR = Q₃ - Q₁
IQR = 6 - 4
IQR = 2.
Answer by
Lanuel
The results of those seven rolls that are shown shows that the range is 5.
What is range?The range is the size of the smallest interval which contains all the data and provides an indication of statistical dispersion. It is measured in the same units as the data.
Given that a six-sided die is rolled 7 times, to give 1, 4, 4, 4, 4, 6, 5, we need to find the range,
The numbers that are gotten include:
1, 4, 4, 4, 4, 6, 5.
It should be noted that range is calculated as:
R = Highest value - Lowest value
R = 6 - 1
R= 5
Therefore, the range from the numbers given is 5.
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Use the formula for computing future value using compound interest to
determine the value of an account at the end of 4 years if a principal
amount of $2,500 is deposited in an account at an annual interest rate of
5% and the interest is compounded daily. (Assume there are 365 days in
a year.)
The amount after 4 years will be $
(Round to the nearest cent as needed.)
The Total amount after 4 years will be $3000 and interest will be $500.
The rate at which you borrow or lend money is called the simple interest. If a borrower takes money from a lender, an extra amount of money is paid back to the lender. The borrowed money which is given for a specific period is called the principal. The extra amount which is paid back to the lender for using the money is called the interest.
You calculate the simple interest by multiplying the principal amount by the number of periods and the interest rate. Simple interest does not compound, and you don’t have to pay interest on interest. In simple interest, the payment applies to the month’s interest, and the remainder of the payment will reduce the principal amount.
The simple interest calculation's mathematical formula:
A = P (1+rt)
P = Principal Amount
R = Rate of interest
t = Number of years
A = Total accrued amount (Both principal and the interest)
Interest = A – P.
Total amount after 4 years
⇒ $2,500 (1+5%×4)
⇒ $2,500 ×1.2\
⇒ $3000
interest= $3000 - $2,500= $500.
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10. You're conducting a vertical jump test on your athletes. This individual just broke the school record
(woohoo!) and achieved a height of 40 inches (1.016m). Calculate their 1) initial velocity and 2) flight time.
1.016m
The jump of the athlete to 40 inches, (1.016 m) is obtained using an initial velocity and time calculated as follows;
1) The initial velocity of the athlete is approximately 4.475 m/s
2) The time of flight which is the time to reach 40 inches and back is approximately 0.91 seconds
Which equation can be used to calculate the velocity and time of the motion?The height achieved by the athlete, h = 40 inches
40 inches = 1.016 mRequired;
1) The initial velocity
Solution;
From the kinematic equation of motion, for a free fall, we have;
v² = u² - 2•g•hWhere;
v = The velocity at the final point = 0 (The athlete stops upward motion at the 1.016 m height and starts downward motion
u = The initial velocity
g = Acceleration due to gravity ≈ 9.81 m/s²
h = The height = 1.016 m
Which gives;
0² = u² - 2 × 9.81 × 1.016
u² = 2 × 9.81 × 1.016 ≈ 19.93392
u = √(19.93392) ≈ 4.465
The initial velocity of the athlete, u, is approximately 4.465 m/s2) The flight time;
The flight time can be found using the equation;
v = u - g•tWhich gives;
t = (u - v)/g
Where;
t = The time taken to reach maximum height.
Therefore;
t ≈ (4.465 - 0)/9.81 ≈ 0.455
The time taken to reach the record height is the same as the time to land back on the ground, which gives;
Time of flight = t + t = 2•t
The time of flight ≈ 2 × 0.445 ≈ 0.910
The time of flight is approximately 0.91 seconds.
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Given m \| nm∥n, find the value of x.
m
n
t
(x+5)°
(x-13)°
Answer:
5
Step-by-step explanation:
564 is divided by 3 = 4
Parallelogram+RSTU+is+rotated+180°+clockwise+about+the+origin.+How+many+pairs+of+parallel+sides+ Does the rotated figure have
RSTU ≅ R'S'T'U' and R'S'T'U' is also a parallelogram with two pairs of parallel sides.
Given that, parallelogram RSTU is rotated 180° clockwise about the origin.
We need to find how many pairs of parallel sides the rotated figure has.
What is the rotation of 180°?Rotation of a point through 180°, about the origin when a point M (h, k) is rotated about the origin O through 180° in an anticlockwise or clockwise direction, it takes the new position M' (-h, -k).
The given transformation was performed on parallelogram RSTU = 180° clockwise rotation.
Given that a rotation is a form of rigid transformation, the shape and size of the preimage RSTU will be equal to the shape and size of the image R'S'T'U'.
Therefore, RSTU ≅ R'S'T'U' and R'S'T'U' is also a parallelogram with two pairs of parallel sides.
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Your question is incomplete, probably the complete question/missing part is:
Parallelogram RSTU is rotated 180° clockwise about the origin. How many pairs of parallel sides does the rotated figure have?
How do I simplify this?
Answer:
Hello,
Step-by-step explanation:
[tex]\dfrac{1}{\sqrt{1} +\sqrt{2} } =\dfrac{\sqrt{2} -\sqrt{1}}{(\sqrt{2} +\sqrt{1})*(\sqrt{2} -\sqrt{1}) } =\dfrac{\sqrt{2} -1}{2-1} =\sqrt{2} -1\\\\\dfrac{1}{\sqrt{2} +\sqrt{3} } =\dfrac{\sqrt{3} -\sqrt{2}}{(\sqrt{3} +\sqrt{2})*(\sqrt{3} -\sqrt{2}) } =\dfrac{\sqrt{3} -\sqrt{2}}{3-2} =\sqrt{3} -\sqrt{2}\\\\\\\dfrac{1}{\sqrt{n-1} +\sqrt{n} } =\dfrac{\sqrt{n} -\sqrt{n-1}}{(\sqrt{n} +\sqrt{n-1})*(\sqrt{n} -\sqrt{n-1}) }=\sqrt{n} -\sqrt{n-1}\\\\[/tex]
[tex]\\\dfrac{1}{\sqrt{1} +\sqrt{2} } +\dfrac{1}{\sqrt{2} +\sqrt{3} } +...+\dfrac{1}{\sqrt{n-1} +\sqrt{n} } =\boxed{\sqrt{n} -1}\\[/tex]
Help me out!!!
Is due today!!
The radius of the large circle would just be the diameter of one of the smaller circles, so 14 inches.
In order to solve this, we will need to find the area of the large circle and subtract the area of two smaller circles.
I'll be using pi as 3.14 for this.
Large Circle:
[tex]A = \pi r^{2}\\A = \pi (14)^{2}\\A = 196\pi \\A = 615.44[/tex]
Smaller Circle:
[tex]A = \pi r^{2}\\A = \pi (7)^{2}\\A = 49\pi \\A = 153.86[/tex]
Now we just simply subtract them.
615.44 - (2 * 153.86)
= 307.72 inches
You might've noticed that is exactly half of the area of the larger circle.
That's because if you halve the radius of a circle, you will get one fourth of original area.
At a local concert hall, the owner charges $7 per ticket. The owner must pay his band
$1200. Every customer receives a brochure that costs the owner $2 for each person.
The number of people that must attend for him to break even
is 240.
Given that the owner charges $7 per ticket, every customer receives a brochure that costs the owner $2 for each person and the owner must pay his band $1200.
We are required to find the people must attend for him to break-even.
Break even point is the point at which there is not any profits or losses.
We have to first form the equation for the break even.
Suppose the number of people be x.
Cost to the owner will be 1200+2x.
Revenue will be 7x.
Break even point will be at the point where 7x=1200+2x
7x-2x=1200
5x=1200
x=240
Hence the number of people that must attend for him to break even
is 240.
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Question is incomplete. It should include the following line:
Calculate the break even level for the owner of local concert hall.
what is the measure of angle COW?
Answer: c: 52
Step-by-step explanation:
the angles are congruent as shown by the hashmarks in the arks so you just need to double it
Answer:
c. 52°
Step-by-step explanation:
Since part of the angle is given (26°) and both parts of the angle are equal. 26° + 26° = 52°
Writing a multi-step equation for a real-world situation
TechWiz Electronics makes a profit of $35 for each MP3 player sold and $18 for each DVD player sold. Last week, TechWiz sold a combined total of 118 MP3 and DVD players.
Let x be the number of MP3 players TechWiz sold last week. Write an expression for the combined total profit (in dollars) TechWiz made from MP3 and DVD players last week.
total profit (in dollars)
The required expression is 17x + 2124 which represents the combined total profit (in dollars) TechWiz made from MP3 and DVD players last week.
What is the algebraic expression?Algebraic expressions are mathematical statements with a minimum of two terms containing variables or numbers.
What are Arithmetic operations?Arithmetic operations can also be specified by adding, subtracting, dividing, and multiplying built-in functions.
Let the number of MP3 players TechWiz sold last week would be x.
Here TechWiz Electronics makes a profit of $35 for each MP3 player sold.
Last week, TechWiz sold a combined total of 118 MP3 and DVD players.
As per the condition,
So the required expression for the combined total profit would be:
= 35x + 18(118-x)
= 35x + 2124 - 18x
= 17x + 2124
Hence, the required expression is 17x + 2124 which represents the combined total profit (in dollars) TechWiz made from MP3 and DVD players last week.
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