Answers
Answer:
[tex]\boxed{x^7+14x^6y+84x^5y^2+280x^4y^3+560x^3y^4+672x^2y^5+448xy^6+128y^7}[/tex]
Step-by-step explanation:
To answer this question, you can use a number pattern that is well-known in algebra & higher mathematics called Pascal's Triangle. However, for simplification reasons, I will use the Binomial Theorem as it is quite difficult to create Pascal's Triangle without just attaching an image. A binomial is any two-term expression that are not like terms (for instance, both are integers that lack a variable).
Part I - Expanding the Binomial using the Binomial TheoremAfter constructing Pascal's Triangle, you can use this method to expand a binomial. To do this, it will require a lot of patience and a lot of space.
Using Pascal's Triangle, we can expand the binomial entirely by referencing the exponents given and the coefficients that we have now solved for using Pascal's Triangle.
Referring to the seventh exponent row of the triangle, we can expand the binomial using a formula referred to as the Binomial Theorem.
The binomial theorem states that:
[tex]\large\boxed{(a+b)^{n}=\sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^{k}, \textsf{where} \binom{n}{k}=\frac{n!}{(n-k)!k!} \textsf {\: and \:} n! = 1 * 2 * 3... \textsf{n}}[/tex]
Therefore, we can use this information to begin defining values and expanding the binomial.
We are given an a, b, and n value. These are defined as a = 2y, b = x, and n = 7.
Therefore, [tex](x + 2y)^{7}=\sum_{k=0}^{7} \binom{7}{k}(2y)^{7-k}(x)^{k}[/tex].
Using this information, you can then solve for every k-value. This will be slightly extensive. To solve for each k-value, ignore the summation sign and everything before it. You only need the information that follows the summation sign: [tex]\binom{7}{k}(2y)^{7-k}(x)^{k}[/tex]. This will then extend and allow you to only use the portions of the equation that require factorials.
Be sure that when you get a final answer that you always apply the exponent (k) to the final result.
k = 0
[tex]\binom{7}{0}(2y)^{7-0}(x)^0 = \frac{7!}{(7-0)!0!}(2y)^{7}(x)^{0}\\[/tex]
[tex]\frac{7*6*5*4*3*2*1}{(7*6*5*4*3*2*1)}(2y)^{7}(x)^{0}[/tex]
[tex]1(2y)^{7} = 2y^7=(2y*2y*2y*2y*2y*2y*2y)= 128y^7[/tex]
[tex]\boxed{128y^7}[/tex]
k = 1
[tex]\binom{7}{1}(2y)^{7-1}(x)^{1}=\frac{7!}{(7-1)!1!}(2y)^{6}(x)^1[/tex]
[tex]\frac{7*6*5*4*3*2*1}{6*5*4*3*2*1} (2y)^{6}(x)^{1}[/tex]
[tex]7(2y)^{6}(x)^{1}=448xy^6[/tex]
[tex]\boxed{448xy^6}[/tex]
k = 2
[tex]\binom{7}{2}(2y)^{7-2}(x)^{2}=\frac{7!}{(7-2)!2!}(2y)^{5}(x)^2[/tex]
[tex]\frac{7*6*5*4*3*2*1}{(5*4*3*2*1)(2*1)}(2y)^{5}(x)^2[/tex]
[tex]\frac{42}{2}(2y)^{5}(x)^2[/tex]
[tex]21(2y)^{5}(x)^{2}=672x^2y^5[/tex]
[tex]\boxed{672x^2y^5}[/tex]
k = 3
[tex]\binom{7}{3}(2y)^{7-3}(x)^3=\frac{7!}{(7-3)!3!}(2y)^{4}(x)^{3}[/tex]
[tex]\frac{7*6*5*4*3*2*1}{(4*3*2*1)(3*2*1)}(2y)^4(x)^3[/tex]
[tex]\frac{210}{6}(2y)^4(x)^3[/tex]
[tex]35(2y)^4(x)^3=560x^3y^4[/tex]
[tex]\boxed{560x^3y^4}[/tex]
k = 4
[tex]\binom{7}{4}(2y)^{7-4}(x)^4 = \frac{7!}{(7-4)!4!}(2y)^3(x)^4[/tex]
[tex]\frac{7*6*5*4*3*2*1}{(3*2*1)(4*3*2*1)}(2y)^3(x)^4[/tex]
[tex]\frac{840}{24}(2y)^3(x)^4[/tex]
[tex]35(2y)^3(x)^4=280x^4y^3[/tex]
[tex]\boxed{280x^4y^3}[/tex]
k = 5
[tex]\binom{7}{5}(2y)^{7-5}(x)^5=\frac{7!}{(7-5)!5!}(2y)^2(x)^5[/tex]
[tex]\frac{7*6*5*4*3*2*1}{(2*1)(5*4*3*2*1)}(2y)^2(x)^5[/tex]
[tex]\frac{2520}{120}(2y)^2(x)^5[/tex]
[tex]21(2y)^2(x)^5 = 84x^5y^2[/tex]
[tex]\boxed{84x^5y^2}[/tex]
k = 6
[tex]\binom{7}{6}(2y)^{7-6}(x)^6=\frac{7!}{(7-5)!6!}(2y)^1(x)^6[/tex]
[tex]\frac{7*6*5*4*3*2*1}{(1)(6*5*4*3*2*1)}(2y)^1(x)^6[/tex]
[tex]\frac{5040}{720}(2y)^1(x)^6[/tex]
[tex]7(2y)^1(x)^6=14x^6y[/tex]
[tex]\boxed{14x^6y}[/tex]
k = 7
[tex]\binom{7}{7}(2y)^{7-7}(x)^7=\frac{7!}{(7-6)(7!)}(2y)^0(x)^7[/tex]
[tex]\frac{7*6*5*4*3*2*1}{(1)(7*6*5*4*3*2*1)}(2y)^0(x)^7[/tex]
[tex]\frac{5040}{5040}(2y)^0(x)^7[/tex]
[tex]1(2y)^0(x)^7=x^7[/tex]
[tex]\boxed{x^7}[/tex]
Part II - Adding/Subtracting all Terms TogetherNow, we have to take a look at the original binomial and discover whether or not we are to add or subtract these terms together. The original binomial was [tex](x+2y)^7[/tex], so we will add (that is the operation of the binomial).
Therefore, we will add all of the terms that we started with following algebraic notation (that is, alphabetically and highest exponent to lowest exponent).
[tex]\boxed{x^7+14x^6y+84x^5y^2+280x^4y^3+560x^3y^4+672x^2y^5+448xy^6+128y^7}[/tex]
This is the final expanded binomial (now a polynomial).
Related Questions
The length of a rectangle is six times its width.
If the perimeter of the rectangle is 112 ft, find its area.
Answers
Answer:
Area = 48*8=384 sq. in
Step-by-step explanation:
Perimeter = 112
Length 6x
Width =x cm
Perimeter = 2(6x+x)
Perimeter = 14x
112 = 14x x=
8 in Width
Length = 6*8=48 in
Area = 48*8=384 sq. in
Did this kinda fast but I hope it helped XD
Length of the rectangle = 48ft
Width of the rectangle = 8ft
Area of rectangle = 384 ft²
What is rectangle?Rectangle is a shape with four straight sides and four angles of 90 degrees (right angles). Two of the sides are longer than the other two.
Given,
Length of a rectangle is six times its width
Let width is x then length is 6x
Perimeter of rectangle = 112 ft
Perimeter of rectangle = 2(Length + Width)
2(Length + Width) = 112
2(x + 6x) = 112
7x = 56
x = 8
Width = 8 ft
Length = 6 × 8 = 48 ft
Area of rectangle = length × width
= 48 × 8
= 384 ft²
Hence, 8ft is width and 48ft is length of the rectangle. 384 ft² is the area of rectangle.
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A point is chosen randomly on AD. What is the probability that the point is on BC?
Answers
What is the length of the line?
There is also D which is square root of 37
Answers
the answer is A
Your Welcome.
Answer:
a
Step-by-step explanation:
1. Is the following quadratic equation in standard form?
y = – 3x^2+ 2x - 4
No
Yes
Answers
Answer:
Yes this equation is in standard form.
Answer:
yes this equation is in standard form
X = ________ ft. (No guessing please)
Answers
Answer:
x=12
Step-by-step explanation:
12(x+8)= 8(30)
Distribute
12x+ 96= 240
Subtract 96 on both sides
12x= 144
Divide 12 on both sides
x=12
What type of triangle is shown below?
isosceles triangle
scalene triangle
right triangle
obtuse triangle
Answers
Answer:
scalene
Step-by-step explanation:
Its a scalene triangle
Solve: Negative one-half + c = StartFraction 31 over 4 EndFraction
c = StartFraction 33 over 4 EndFraction
c = 8
c = StartFraction 29 over 4 EndFraction
c = 7
Answers
Answer:
A
Step-by-step explanation:
i am a bakon trust me
plz brainliest for bakon :D i know u want B)
Answer:
A
Step-by-step explanation:
got it right on edge so......
What is the area of the trapezoid?
Answers
Answer:
The answer is 36 squared centimeters
Solve for K: 40 + K = 56
K=
Answers
Answer:
k=16
Step-by-step explanation:
To solve this, we can isolate k.
40+k=56.
We can do this by subracting 40 from both sides.
40+k-40=k
56-40=16
k=16
First isolate or get k by itself by subtracting 40 from both sides.
That leaves us with k = 16.
To check our answer, substitute 16 back into the equation.
So we have 40 + (16) = 56 or 56 = 56.
Add and simplify.
X+2x/x+1
Answers
Answer:
x+3
Step-by-step explanation:
x+2x1+1
x+2+1
x+3
Answer:
x+3
Step-by-step explanation:
x+ 2x /x+1
= x2+3x /x
=x+3
What is the mode of 4, 9, 9, 9, 4, 6, 6, 6, 9?
Answers
Answer:
9
Step-by-step explanation:
That's the most common number there
Answer:
9
Step-by-step explanation:
The mode of a set of data values is the value that appears most often.
in this set, 9 appears 4 times -> 9 is mode
Jake is wrapping a gift that is in a box with the following dimensions.
Length: 12 inches
Width: 4 inches
Height: 8 inches
Havana is wrapping a gift whose box's dimensions are all twice the length of the lengths in Jake's box. How many times larger is the surface area of Havana's box than Jake's box?
Answers
Answer:
Because you are looking for square inches of wrapping paper, you are looking for the surface area of the box. A three-dimensional box has a top and bottom that are the same size, a front and back that are the same size, and 2 sides that are the same size...or SA = 2(top/bottom area) + 2(front/back area) + 2(side/side area). Our equation fills in nicely like this:
SA = 2(6 * 14) + 2(6 * 8) + 2(14 * 8)
SA = 2(84) + 2(48) + 2(112)
SA = 168 + 96 + 224
SA = 488 square inches of paper to wrap the box.
Step-by-step explanation:
A. 11 square centimetres
B. 22 square centimetres
C. 15 square centimetres
D. 30 square centimetres
Answers
Step by step: 5x6=30
Factor the polynomial completely.
x^2 + x + 3
Answers
Answer:
ehswFYDHSDDSHJVBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
Step-by-step explanation:
In the 30-60-90 triangle below, side s has a length of __and side q has a
length of
Answers
Answer:
Step-by-step explanation:
We are given a 30-60-90 triangle, where the length of the hypotenuse is 10 units. We are to find the length of sides s and q. Therefore, s = 5 units and q = 5√3 units. In the given 30-60-90 triangle, side s has a length of 5 units and side q has a length of 5√3 units.
Solve x + x + 2 + 2x = -2.
1
-1
0
-2
Answers
Answer:
[tex]x=-1[/tex]
Step-by-step explanation:
[tex]x + x + 2 + 2x = -2[/tex]
First, we need to collect like terms:
[tex]4x+2=-2\\4x=-4[/tex]
Then we divide both sides by 4:
[tex]\frac{4x}{4}=\frac{-4}{4}\\x=-1[/tex]
The answer is x = -1.
To solve this equation, you must collect like-terms by adding together x terms and numbers, and then rearrange to get x on one side and numbers on the other:
x + x + 2 + 2x = -2
4x + 2 = -2
- 2
4x = -4
÷ 4
x = -1
I hope this helps! Let me know if you have any questions :)
the isoceles triangle has a base that measures 14 units.the value of y, the length of each leg, must be
Answers
Answer:
greater than 7
Step-by-step explanation:
Factor the expression completely.
10x5 + 18
I
Answer:
Submit Answer
E
BI
Answers
Factor the expression completely.
10 × 5 + 18
50 + 18
68
Answer:
[tex]68[/tex]
Step-by-step explanation:
[tex]10 \times 5 + 18 \\ 50 + 18 \\ = 68[/tex]
hope this helps
brainliest appreciated
good luck! have a nice day!
A train moves at a constant speed of 8 miles every 6 minutes. How far will the train in 3 minutes, 15 minutes, and 60 minutes.
Answers
Answer: If the train moves a constant speed of 8 miles for every 6 minutes then if the time where to be 3 minutes you would have to divided 2 and 8 because the time was split in half you would also have to split the 8 as well. Now if the time was 15 minutes you would have to add more since 6 got 9 more you will have to add 9 more to 8.
6 minutes = 8 miles
3 minutes = 4 miles
15 minutes = 17 miles
Hope this helps you :3.
The train will move 4 miles, 20 miles and 80 miles in 3 minutes, 15 minutes and 60 minutes respectively.
What is multiplication?Multiplication is mathematical arithmetic operation. It is also a process of adding same types of expression to some number of times.
Example - 2 × 3 means 2 is added three times or 3 is added 2 times.
Given, a train moves at a constant speed of 8 miles every 6 minutes.
That means in one minute,
= total number of miles / total time taken
= 8 /6
= 4/3 miles
Now, for 3 minutes means 3 times 4/3
= 3 x 4/3
= 4 miles
For 15 minutes,
= 15 x 4/3
= 20 miles
For 60 minute
= 60 x 4/3
= 80 miles.
Therefore, the train will move 4 miles, 20 miles and 80 miles in 3 minutes, 15 minutes and 60 minutes respectively.
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There are 1 1/2 pounds of bricks in a bag. Each brick weighs 1/6 of a pound. How many bricks are in the bag?
Write your answer as a fraction or as a whole or mixed number.
Answers
Answer:
answer = about 9
Step-by-step explanation:
1 1/2= 3/2. 3/2 divided by 1/6= 8.99999999999999999 or about 9.
A school newspaper took a survey of 100 students. The results of the survey showed that 43 students are
fans of the Buffalo Bills, 27 students are fans of the New York Jets, and 48 students do not like either
team. How many of the students surveyed are fans of both th
Answers
Triangles L M N and L K N are connected at side L N. A line is drawn from point M to point K and intersects side L N at point P. Which additional information, if true, would help to prove that ΔLMP ≅ ΔNMP by HL? Select two options. Point P is the midpoint of MK. Line MK is the perpendicular bisector of LN. ML ≅ MP ML ≅ MN PK ≅ PK
Answers
Answer:
Line MK is the perpendicular bisector of LN. and ML is congruent to MN
Step-by-step explanation:
Answer: Line MK is the perpindicular bisector of LN.
and ML is ≅ MN.
Step-by-step explanation:
I received a perfect score on edge
(QUICK ANSWERS) Which expression is equivalent to -3(2m-1)- n
Answers
Answer:
Choose the fourth option
Step-by-step explanation:
This is a simple distribution.
[tex]-3(2m-1)-n\\-6m+3-n\\-6m-n+3[/tex]
It's the last option.
Answer:
D. -6m-n+3
Step-by-step explanation:HOPES THIS HELPS!
Explain how to use a graph of the function f(x) to
find f(3).
Answers
Answer:
For f(3), the input is 3 and you are looking for the output. To determine the function's value when x = 3, go to the value of 3 on the x-axis and then locate the graph for that value of x. Determine the value of y from the y-axis at that location on the graph.
Step-by-step explanation:
up there
Anyone know the answer to this ?
Answers
Answer:
Triangle HBN ~ triangle LYR
=> HN/LR = HB/LY
=> (3x - 7)/(x + 11) = 28/21
=> 21(3x - 7) = 28(x + 11)
=> 63x -147 = 28x + 308
=> 35x = 455
=> x = 455/35 = 13
There is no answer to select. But I still believe there is no mistake in my calculation.
Hope this helps!
:)
Find the missing endpoint if the midpoint is at (2,-5) and one endpoint is
at (12,-5).
Answers
Answer:
(-8, -5)
hope this helps you
..........
Plzz help me its is math
Answers
Answer:
A
Step-by-step explanation:
Whenever you square a real number, the result will always be positive, because even if you are squaring a negative number a negative times a negative is a positive. However, when you cube a negative number, the result CAN be negative, because you are multiplying a negative number by itself three times. Therefore, the answer is only A. Hope this helps!
Seth wants to cover the top of a small table with glass parallelogram tiles. The top of the table measures 18 inches by 14 inches. If each tile has four sides of length 3 inches with a height of 2 inches, about how many tiles will he need? 26 42 63 84
Answers
Answer:
42
Step-by-step explanation:
edge 2022 :) I hope I helped a bit credit to the comment above me
The number of tiles he will need 42.
What is the area of a parallelogram?The area of a parallelogram is the product of the base and the height of the figure.
The area of a parallelogram = Base x Height
We are given that top of the table measures 18 inches by 14 inches. If each tile has four sides of length 3 inches with a height of 2 inches.
Area of table = Base x Height
= 18 x 14
= 252
Area of glass parallelogram tiles = Length x width
= 3 x 2
= 6
Now,
252 = 6 x number of tiles
number of tiles = 252 / 6
= 42
Learn more about the area;
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Given: CD = I, and
matrix inverse
Find the elements of D.
a =
b =
c =
d =
e =
f =
g =
h =
i =
Answers
Answer:
Edge answers in order:
a= -9
b= 7
c= 12
d= -5
e= 4
f= 7
g= 3
h= -2
i= -4
Step-by-step explanation:
just did the activity on edge.
Answer:
a = -9
b = 7
c = 12
d = -5
e = 4
f = 7
g = 3
h = -2
i = -4
Step-by-step explanation:
(Question) Write to Explain; Shauna evaluated the expression 9 - 6 + (12/3) *2 and got the answer 4 is she correct? Why or why not?
And if this helps I'll add the screenshot I took of it.
Why is she not correct or Why is she correct.
Answers
the answer is, no.. she is not correct.
she solved the equation/question incorrectly.
why? Shauna subtracted 9 - 12/3 = 5
she multiplied 5 and 2 = 10
then subtracted by 6.
so she got 4.
but, this how you do it.
subtract 9 and 6, equaling 3.
12/3 x 2= 8
add 8 and 3, which equals 11.
the correct answer is 11!
