The binomial theorem says
[tex]\displaystyle (a + b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k \\\\(a+b)^n= \binom n0 a^n + \binom n1 a^{n-1}b + \cdots + \binom n{n-1} ab^{n-1} + \binom nn b^n[/tex]
where
[tex]\dbinom nk = \dfrac{n!}{k!(n-k)!}[/tex]
is the so-called binomial coefficient. The binomial coefficients follow a neat pattern called Pascal's triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
where the n-th row, starting with n = 0, lists the coefficient of the k-th term in the sum (with 0 ≤ k ≤ n). For example,
[tex]\displaystyle n=0 \implies (a + b)^0 = \sum_{k=0}^0 \binom nk a^{0-k} b^k = \binom00 a^{0-0} b^0 = \underline{1}[/tex]
[tex]\displaystyle n=1 \implies (a + b)^1 = \sum_{k=0}^1 \binom 1k a^{1-k} b^k = \binom10 a^{1-0} b^0 + \binom11 a^{1-1}b^1 = \underline{1}a + \underline{1}b[/tex]
[tex]\displaystyle n=2 \implies (a + b)^2 = \sum_{k=0}^2 \binom 2k a^{2-k} b^k = \binom20a^{2-0}b^0+\binom21a^{2-1}b^1+\binom22a^{2-2}b^2 = \underline{1}a^2 + \underline{2}ab+\underline{1}b^2[/tex]
Each row starts and ends with 1, and every coefficient in between is obtained by adding the coefficients directly above and to the left. The next row, for instance, would be
1 1 + 4 4 + 6 6 + 4 4 + 1 1
or
1 5 10 10 5 1
which is to say,
[tex]\displaystyle (a + b)^5 = \underline{1}a^5+\underline{5}a^4b+\underline{10}a^3b^2+\underline{10}a^2b^3+\underline{5}ab^4+\underline{1}b^5[/tex]
In your case, we have (d - 4b)², so we take a, b, and n above with d, -4b, and 2, respectively:
[tex]\displaystyle (d - 4b)^2 = \sum_{k=0}^2 \binom 2k d^{2-k} (-4b)^k[/tex]
[tex]\displaystyle (d - 4b)^2 = \sum_{k=0}^2 \binom 2k (-4)^k d^{2-k} b^k[/tex]
[tex]\displaystyle (d - 4b)^2 = \binom20 (-4)^0 d^{2-0} b^0 + \binom21 (-4)^1 d^{2-1} b^1 + \binom22 (-4)^2 d^{2-2} b^2[/tex]
[tex]\displaystyle (d - 4b)^2 = 1\cdot1\cdot d^2\cdot1 + 2 \cdot (-4) d \cdot b + 1 \cdot 16 \cdot 1 \cdot b^2[/tex]
[tex]\boxed{(d - 4b)^2 = d^2 - 8bd + 16b^2}[/tex]
Pleaseeeeee I need help!
THANK YOU
[tex] \: \: \: \: \: \: \: [/tex]
[tex] \frac{3}{10} \\ \\ or \: in \: alternate \: \: form \\ \\ 0.3[/tex]
refer to the attachment for explanationhope it helpsWhat +1/16 = 3/2 answer please
Answer:
1 7/16
Step-by-step explanation:
What is the radical form of the expression 4 3/2
Which expression has the greatest value when c = 7? 12 - 2C 0R C - 5
please show working
If c = 7, then
12 - 2c
= 12 - 2(7)
= 12 - 14
= -2
c - 5
= (7) - 5
= 2
=》+2 > -2.
So, c - 5 has greater value than 12 - 2c, if c = 7.
______
Hope it helps ⚜
a homeowners association charges a 4.65 fee for each day your payment is late if a bill was originally 103.55 and paid 22 days late then how much is owed to the home owners?
Answer:
207.1
Step-by-step explanation:
4.65×22=102.3
103.55+102.3= 207.1
Write the equation of the line in fully simplified slope-intercept form.
Answer:
y=5/6x+5
Step-by-step explanation:
Just ask!
Hope this helps.
Sorry if I'm wrong.
~Happy Holidays!!~ o(* ̄▽ ̄*)ブ
What is the solution for -10 < x - 9?
x > -19
x < -1
x < -19
x > -1
Answer:
x > -1
Step-by-step explanation:
Add 9 to both sides of the inequality.
-10 +9 < x -9 +9
-1 < x
In the form of the answers, ...
x > -1
The solution for the given inequality is x > -1. The correct option is the last option x > -1
Linear inequalitiesFrom the question, we are to determine the solution for the given inequality
The given inequality is -10 < x - 9
To solve an inequality, we will solve for the variable. The variable here is x
The inequality can be solved as follows
-10 < x - 9
Add 9 to both sides
-10 +9 < x - 9 + 9
Simplifying
-1 < x
Now, if -1 < x
Then,
x > -1
Hence, the solution for the given inequality is x > -1. The correct option is the last option x > -1
Learn more on solving Linear inequalities here: https://brainly.com/question/246993
1cm
1
6 cm
Diagram NOT
accurately drawn
Scm
11 cm
9 cm
| 6cm
18 cm
The diagram shows a shape.
All of its corners are right angles.
Work out the area of the shape.
Answer:
110cm²
Step-by-step explanation:
*Diagram is ALSO not accurately drawn
18cm-6cm-5cm=7cm
11cm-9cm=2cm
Area:
11cm×6cm=66cm²
7cm×2cm=14cm²
6cm×5cm=30cm²
Area of the shape:
66cm²+14cm²+30cm²=110cm²
Can you help me i Don’t understand
Answer:
A. {-1, 2, 7}
Step-by-step explanation:
For a set of ordered pairs, the domain is the set of first-numbers (x). The range is the set of second numbers (y, or g(x)).
The first step here is to find the g(x) values for each of the listed x-values.
The attachment shows a table.
Often, we like to list the domain and/or range values in numerical order, eliminating any duplicates. That is, the set of range values {2, -1, -1, 7} would be listed as ...
{-1, 2, 7} . . . . . . matches choice A
_____
You find the g(x) values by substituting for x and doing the arithmetic.
g(-2) = (-2)² -2 = 4 -2 = 2, for example
PLEASE HELP!!!
A golfer hits a ball with an initial velocity of 32.7 m/s from the ground. Find the following (round all answers to 2 decimal places...no units):
When does the ball hit the ground? ___
What is the highest height the ball goes? ___
When does the ball reach its max height? ___
How high is the ball after 4.3 seconds? ___
When is the ball 24 meters off the ground? ___ and ___ (enter the smaller value first)
Answer: 1. it doesnt since the velocity is so high, it will land in the water.
2. the ball goes about 500 feet in the air.
3. the ball reaches its max height when it starts to go down
4. the ball is at 250 since it takes 8.6 seconds for it to reach 500 feet
Step-by-step explanation:
To repair a roof that is 4 meters high Mr. Jeff leans a 5-meter ladder against the side of the building. To reach the roof how far away from the building should he place the base of the ladder?
hi have a good day.
I think correct answer is-3
A rectangle has an area of 35x + 42 square units. If the
width of the rectangle is 7 units, what is the length of the
rectangle?
Answer:
length = 5x + 6
Step-by-step explanation:
35x + 42 = 7*L since area of a rectangle is length * width
(35x + 42) divide by 7 = 7L divide by 7
5x +6 = y
What is 8,024.62 in words?
Answer:
eight thousand twenty four sixty two hundreths
Step-by-step explanation:
you can replace the decimal pint with an and so that its
eight thousand twenty four and sixty two hundreths
Answer: eight thousand twenty-four and sixty-two hundredths
Step-by-step explanation:
What is the length of the longest side of an isosceles triangle with leg lengths 2x+12 and 5x-9 and base length 6x-2?
1/3x^2 = -3
The solution to the equation is ___?
A) x = 3
B) x = -3
C) x = ±3
D) no solution
[tex] \frac{1}{3} {x}^{2} = - 3 \\ = > {x}^{2} = - 3 \times 3 \\ = > {x}^{2} = - 9 \\ = > {x} = \sqrt{ - 9} \\ = > x = ±3[/tex]
Answer:
C) x = ±3
Hope you could get an idea from here.
Doubt clarification - use comment section.
Factorize
A) [tex]x^{2} -x-132[/tex]
B) [tex]9t^{2} -u^{2}[/tex]
C) [tex]4p^{2} -9[/tex]
D)[tex]162-8t^{2}[/tex]
E) [tex]p^{2} x-4q^{2} x[/tex]
i dont know how to do this stuff
im choosing brainliest
Answer:
5 - 15 = -10
Step-by-step explanation:
Answer:
5 - 15 = -10
Step-by-step explanation:
If you start with 5 and then subtract 15, you end up at -10.
10 is 20% of (blank)
Answer:
50
Step-by-step explanation:
Find the slope given the the two points
Answer:
the answer is 2
Step-by-step explanation:
use the slope equation
y^1-y^2/x^1-x^2
-1-(-5)/1-(-1)
4/2
2
9(2x129) = ?
Please help me im suffering math is hard
Answer:
18x^129
Step-by-step explanation:
Answer: That would be 2,322!
Step-by-step explanation:
A line graph titled If the election were held today, whom would you vote for? It shows Weeks until election on the x axis from 6 to 1 and percentage of voters on the y axis. Line labeled Diaz starts at 34 on 6, to 30 on 4, 34 on 3, 35 on 2, 38 on 1. Line labeled Smith starts at 32 on 6, to 35 on 4, 34 on 3, 33 on 2, 34 on 1. How can media coverage of polling influence the outcome of an election? Check all that apply. Voters are less informed about the potential outcome of the election. Candidates learn more about their standing. Voters may be more or less motivated to support a candidate. Candidates are less informed about voter desires. Campaigns are more able to predict election outcomes.
Answer:
2,3,5 or B,C,E
Step-by-step explanation:
Probs too late but I got it right on edge. Hope this helps anyone else :)
Please solve on same Kind of coordinate plane. Thank you!
Answer:
(0,8)
Step-by-step explanation:
In the table adam recorded the miles he travled each day while traveling from Florida to California. Calculate the average rate of change between day 4 and day 6.
Table:
End of day Distance
1 383
2 682
3 1132
4 1503
5 1906
6 2196
Answers: A) 231 Miles
B) 290 Miles
C) 347 Miles
D) 440 Miles
Hi!
To calculate the rate of change between days 4 and 6, we simply take those points, (4, 1503) and (6, 2196) and use the equation in the picture below:
Now, when we plug it in the equation, it looks like this:
[tex]\frac{2196-1503}{6-4} = \frac{693}{2} = 346.5[/tex]
Therefore, your answer is 346.5 or, rounded, 347.
Meaning, your answer is C
Hope this helps and have a great day! :D
30,098 / 6
Quotient and Remainder
Answer:
5,016.33333333
Step-by-step explanation:
divide
Kate is 11 45
years old. Karina is 1 16
years younger than Kate and Jane is 1 13
years younger than Karina. How old is Jane?
Please show step by step need it now
Answer:
x = 6+3y
y = y (any number)
the rest should follow the same steps
Step-by-step explanation:
2x-6y=12 -> divide both sides by 2 (GCF)
x-3y = 6 -> solve for x, because it has 1 as its coefficient, or the lowest coefficent
x = 6+3y
-5x+15y=-30 -> divide both sides by 5
-1x+3y=-6 -> plug in x from previous equation
-(6+3y) + 3y = -6
-6-3y+3y = -6
y = y
Which value of k makes 5-k + 12 = 16 a true statement? Choose 1 answer. А k = 1 B k = 2 k = 3 k=4
PLEASE HELP ME AND SAVE MY LIFEE
ILYYY
Answer:
option A
K = 1
is the answer.
hope it helps
What is the slope of a line that contains the point
(3,7) and has a y-intercept of 1?
Which expression is equivalent to [(2a-3b4)2(3a5b)-2]-1 ? Assume a =/ 0, b =/ 0
Answer: The Third option
Step-by-step explanation:
1/36a^4b^10
Hope this is right!
The expression equivalent to the given expression is 1/36a⁴b¹⁰. Therefore, option C is the correct answer.
What is an equivalent expression?Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.
The given expression is ((2a⁻³b⁴)²/(3a⁵b)⁻²)⁻¹.
Now, ((2²a⁻⁶b⁸)/(3⁻²a⁻¹⁰b⁻²))⁻¹
= (2⁻²a⁶b⁻⁸)/(3²a¹⁰b²)
= (a⁶)/(3²2²a¹⁰b²b⁸)
= 1/36a⁴b¹⁰
Therefore, option C is the correct answer.
To learn more about an equivalent expression visit:
https://brainly.com/question/28170201.
#SPJ7
How do you work out the problem x+3=4+x
Answer:
No solution
Step-by-step explanation:
Since both x's are positive, there's no way to put to x on either side, because when you transfer sides, you add/subtract the opposite, so there can't be a value for x