Mathematics High School

(a + b)2 = 1a^2 + 2ab + 1b^2
(a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3

How are binomial expansions related to Pascal’s triangle?

Answers

Answer 1
Consider the binomial [tex]\displaystyle{ (a+b)^\displaystyle{n[/tex],

where n=0, 1, 2, 3, ...

For example 

[tex]\displaystyle{ (a+b)^0=1[/tex]

[tex]\displaystyle{(a+b)^1=1a+1b[/tex]

[tex]\displaystyle{(a+b)^2=1a^2+2ab+1b^2[/tex]

[tex]\displaystyle{ (a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3[/tex]
...
...


Consider the Pascal's triangle, as shown in the picture, where the very first row is denoted by row 0, the second by row 1, the third by row 2 and so on...


We notice that the coefficients of the expansion of [tex](a+b)^n[/tex], are the entries in the [tex]n^{th}[/tex] row of Pascal's triangle.


(a + B)2 = 1a^2 + 2ab + 1b^2(a + B)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3 How Are Binomial Expansions Related
Answer 2

Answer:

The coefficients of the terms come from rows of the triangle.

If the exponent is n, look at the entries in row n.

Related Questions

If a = -4, and b = 3 what is |a-b|

Answers

Absolute value is the number of units a number is from 0. Basically, you just make everything positive. For example |-5| is 5; |5| is also 5.

Using this concept:

|a - b|
|-4 - 3|
|-7|
7

Hope this helps!
la-bl
l-4-3l
l-7l
7
I think that’s how you do it

A red kangaroo can cover 40 feet in 1 jump. How many feet can the red kangaroo cover in 12 jumps.

Answers

The red kangaroo can jump 480 feet in 12 jumps. (40 * 12 = 480)

What is the value for X?
- 24
-9
-21
-14
what is the measure of <1?
- 45
- 60
- 120
-125

Answers

180-(5x+15)=3x-3
180-5x-15=3x-3
-5x-3x=-3-180+15
-8x= -168
x= 168/8= 21
x=21

angle <1
180-(3x-3)
180-(3×21-3)
180-(63-3)
180-60= 120°
Final answer:

The value of X is -14 and the measure of <1 is 60 degrees.

Explanation:

The value of X is -14. This can be determined by checking each option and finding the one that satisfies the given condition. Similarly, the measure of <1 is 60 degrees. This can also be determined by checking each option and finding the one that matches the given criteria.

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PLEASE HELP OVER 50 POINTS ONE QUESTION

8. Fill in the missing reasons for the algebraic proof below.
A:
B:
C:

Answers

a) distribution property
b) addition property of equality
c) division property of equality

hope this helps

a. Distributive property of addition
b. Subtraction property of equality
c. Division property of equality

What is the median of the data set: 15, -2, 0, 15, 2, 12

Answers

put in order
-2, 0, 2, 12, 15, 15
median is the middle number
so
median = (2+12)/2 = 14/2 = 7

hope it helps
To find the median, you have to number it from smallest to largest. 
-2,0,2,12,15,15
Since there are 2 medians, you find the average of those two numbers.
2+12 is 14/2 is 7.
7 is the median.

the height of a triangle is 4 times its base. if he are of a triangle is 200cm, what is the height

Answers

A = (h * b) / 2
A = 200
h = 4b

200 = (b(4b) / 2
200 * 2 = b * 4b
400 = 4b^2
400 / 4 = b^2
100 = b^2
sqrt 100 = b
10 = b ..... base is 10 cm

h = 4b
h = 4(10)
h = 40 <=== height is 40 cm

First four terms of f(n)=3n-1

Answers

f(n)=3n-1
First term,
n=1
f(1)=3(1)-1
=2

Second term,
n=2
f(2)=3(2)-1
=5

Third term,
n=3
f(3)=3(3)-1
=8

Fourth term,
n=4
f(4)=3(4)-1
=11

Therefore, the first four terms are 2,5,8,11

The first four terms of the function f(x) = 3n - 1 are 2, 5, 8, and 11.

Given,

f(n) = 3n - 1

We need to find the first four terms.

What is a function?

A function has an input and an output.

Example:

f(x) = x + 1

x = 1,2,3 are inputs.

f(1) = 1 + 4 = 5

f(2) = 2 + 4 = 6

f(3) = 3 + 4 = 7

5, 6, and 7 are the outputs.

We have,

f(n) = 3n - 1

We can assume that the first four terms can be found with n= 1,2,3,4.

f(1) = 3 x 1 - 1 = 3 - 1 = 2

f(2) = 3 x 2 - 1 = 6 - 1 = 5

f(3) = 3 x 3 - 1 = 9 - 1 = 8

f(4) = 3 x 4 - 1 = 12 - 1 = 11

Thus the first four terms of the function f(x) = 3n - 1 are 2, 5, 8, and 11.

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If a six sided die is tossed two times and '3' shows up both times, the probability of a '3' on the third trial is

Answers

1 out of 6 time you would be able to get three the third time 

If a six sided die is tossed two times and '3' shows up both times, the probability of a '3' on the third trial is  [tex]\frac{1}{6}[/tex]

Given :

A six sided die is tossed two times and '3' shows up both times

'3' shows up both the times .

When the dice is rolled third time. there are 6 possibilities comes up

In the third trial , we have six possibilities 1,2,3,4,5,6

Out of six possibilities , the chance of getting '3' is only once

So, the probability of a '3' on the third trial is [tex]\frac{1}{6}[/tex]

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There are 8 pencils in package.how many packages will be needed for 28 children if each child get 4 pencils

Answers

8 pencils = 1 package
28 children need pencils
Each child gets 4 pencils

28x4 = 112 (pencils needed)

112/8 = 14 packages needed.

solve for x
5.1 x = 1.02

Answers

5.1(x) = 1.02

5.1(x)/5.1 = 1.02/5.1

x = 1.02/5.1

x = 0.2

hope this helps
The answer is .02 for this question 

solve 7d + d - 4d - 7 - 5d

Answers

-1d-7 is the answer I'm pretty sure

 1, Simplify  7d+d-4d-77d+d−4d−7 to 4d-74d−7

4d-7=5d4d−7=5d

 2.Subtract 4d4d from both sides
-7=5d-4d−7=5d−4d

 3. Simplify 5d-4d5d−4d to dd
-7=d−7=d

 4. Switch sides
d=-7d=−7
5. so your answer would be -7

1. solve the proprotion k/5=5/125.
a) 1/25
b) 25
c) 5
d) 1/5


2. round the answer of 0.4x-0.2=2.5 to two decimal places.
a) -2.70
b) -6.75
c) -5.75
d) -3.25

3. what is the y-intercept of 4x-3y=-12?
a) -4
b) 3
c) -3
d) 4

Answers

1. solve the proportion k/5=5/125.
First reduce 5/125.
5/125 = 1/25
Now you can cross multiply.
25k = 5
k = 5/25
k = 1/5
d) 1/5

2. round the answer of 0.4x-0.2=2.5 to two decimal places.
0.4x=2.7
x = 6.75
b) -6.75

3. what is the y-intercept of 4x-3y=-12?
The y-intercept is when x = 0.
4(0) - 3y = -12
-3y = -12
y = 4
d) 4

What is the measure of each interior angle for a regular polygon with 30 sides?

Answers

Number of degrees in the 30 sided regular polygon is (30-2)x180 = 5040
5040/30=168 degrees in each interior angle

In how many ways can the letters in the word 'alaska' be arranged?

Answers

The word Alaska can be arranged in 720 ways

What’s he hypotenuse, x, if ( 20, 21,x) is a Pythagorean triple?

Answers

If x is the hypotenuse, that means that sqrt(20^2+21^2)=x=29 since sqrt(a^2+b^2)=c or a^2+b^2=c^2

Answer:

B) - 29

Step-by-step explanation:

Is it possible for two linear functions whose graph are parallel to have the same y-intercept

Answers

If you have e two linear functions and on a graph they are parallel that means they have the same slope with different y intercepts. If they are parallel and have the same y intercept then they would be the same graph. So no it’s not possible because there would not be two different graphs/functions; they would be exactly the same.

The sum of two numbers is 39 . the smaller number is 13 less than the larger number. what are the numbers

Answers

13 and 26

13+26=39 and 26-13=13

The required numbers are given as 26 and 13.

What are equation models?

The equation model is defined as the model of the given situation in the form of an equation using variables and constants.

here,
let the number be x and y,
According to the question,
x + y = 39 - - - - (1)
x =  y + 13 - - - - - (2)

Put value from equation 2, in equation 1
y + y + 13 = 39
2y = 39 - 13
y = 26/y
y = 13

Now,
x = 13 + 13

x = 26

Thus, the required numbers are given as 26 and 13.

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Which number is a solution of the inequality? 6 ≥ 6k?

Answers

This isnt a inequality

What is the average of 12 1/2, 13, and 13?

Answers

To get the average add up the totals and divide by the amount
(12.5+13+13)/3

=38.5/3 
=12.833

The temperature was 8 degrees below zero. What does the 0 represent?

Answers

The zero represents the origin
the zero ( 0 ) is the orgin

i hope this helped! have a wonderful day!!!


Why is it incorrect

Answers

keeping in mind that, an absolute value, is in effect a piece-wise function, so is a duplet, so |whatever here|   is really two siblings, the positive and negative siblings  +(whatever here) and -(whatever here), thus

[tex]\bf \cfrac{|3v-1|}{5}\ge 4\implies |3v-1|\ge 20\implies \begin{cases} +(3v-1)\ge 20\\ -(3v-1)\ge 20 \end{cases}\\\\ -------------------------------\\\\ +(3v-1)\ge 20\implies 3v-1\ge 20\implies 3v\ge 21\implies v\ge \cfrac{21}{3} \\\\\\ \boxed{v\ge 7}\\\\ -------------------------------\\\\[/tex]

[tex]\bf -(3v-1)\ge 20\implies -3v+1\ge 20\implies -3v\ge 19 \\\\\\ \stackrel{\textit{negative division, the sign changes}}{v\le \cfrac{19}{-3}}\implies \boxed{v\le -\cfrac{19}{3}}\\\\ -------------------------------\\\\ 7\le v\le -\cfrac{19}{3}[/tex]

Raul is selling coupon books. He started with 50 coupon books and has already won a prize for selling at least 10 coupon books. The total amount of money Raul will make from selling x coupon books is represented by a function.



f(x)=20xf(x)=20x



What is the practical domain of the function if Raul sells 50 coupon books?

Answers

The practical domain would be all integers 10 to 50.

40% of a certain number is 80 what is that number ?

Answers

let number be x
40/100(x)=80
x=200
Since 40%=80, 80% must be 160, so half of 40 is 20  so the 20% is 40, 40+160=200

Your answer is 200.

Geometry help please

2 attachments, 1 question

Answers

The answer to this question should be: 
1. definition of supplementary angles
2. 115
3. definition of acute angle

Supplementary angles mean that the angle will add up to 180°. In this case, that means PQR + RQS= 180° where PQR=115°
So the RQS would be 180-115=65° Acute angle is when the angle is <90° so RQS is acute angles

Which statement is true about regular exercise? 1.It decreases your resting heart rate.
2.It decreases your lung capacity.
3.It decreases your intelligence.
4.It increases your blood pressure.

Answers

It decreases your resting heart rate

Answer:

got correct on edmentum mastery test, hope this helps

Find the value of a.

Answers

set them equal then solve with basic algebra
a = 18
6a + 11 = 2a + 83
6a - 2a = 83 - 11
4a = 72
  a = 72/4
  a = 18

Factor the expression:
30x^3-5x^2-18x+3

PLZ SHOW YOUR WORK

Answers

Lets split this up

30x^3 -5x^2  turns into 5x^2(6x-1)

-18x+3 turns into -3(6x-1)

we can then combine them to get (5x^2-3) (6x-1)

You are playing a video game. For each diamond you collect, you earn 80 points. You also earn 100 bonus points by completing the level in less than 2 minutes. You need to earn more than 1220 points to continue on to the next level. Write and solve an inequality that represents the numbers dd of diamonds you must collect in less than 2 minutes to advance to the next level.

Answers

Given

For each diamond you collect, you earn 80 point

earn 100 bonus points by completing the level in less than 2 minutes.

earn more than 1220 points to continue on to the next level

Find the inequality that represents the numbers dd of diamonds you must collect in less than 2 minutes to advance to the next level.

To proof

As given in the question

For each diamond  earn point = 80

bonus points by completing the level in less than 2 minutes = 100

earn more than 1220 points to continue on to the next level

Let us assume that the number of diamond = d

The inequality written in the form

we  get

[tex]80d + 100 \geq 1220[/tex]

now solving the above equation

[tex]80d \geq1120[/tex]

[tex]d \geq14[/tex]

Thus the  numbers  of diamonds you must collect in less than 2 minutes to advance to the next level be 14.

Hence proved

Find the values of x and y.

a. x=44, y=66
b. x=46, y=44
c. x=90, y=44
d. x=90, y=46

Answers

m<B = m<C = 46
m<BAC = 180 - 2(46) = 180 - 88
y = 88/2 = 44

x = 180 - (44+46) = 180 - 90 = 90

answer
c. x=90, y=44

[2/(n+2)] - (3/n+2) = 5

Solve by u-substitution

Answers

let n+2=u so, the equation became= [2/u]-[3/u]=5 => [(2+3)/u]=5 => 5/u=5 => u=5/5=1 thus, u=1 we know u=n+2 so, n+2=1 => n=1-2=-1 so, n=-1
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