Question

Anyone good with the Binomial Theorem or Pascal’s triangle?

Answer 1

Mmmm maybe. Sup? Have a question? c:

Answer 2

yea

Answer 3

Expand the following using the Binomial Theorem and Pascal’s triangle.
(x + 2)^6

Answer 4

\[\large\bf\sf (x+2)^6=\text{_}x\cdot 2+\text{_}x\cdot 2+\text{_}x\cdot 2+\text{_}x\cdot 2+\text{_}x\cdot 2+\text{_}x\cdot 2+\text{_}x\cdot 2\]

Mmm ok good, so your x's count DOWN from 6, while your 2's count UP from zero.\[\bf\sf (x+2)^6=\text{_}x^6\cdot 2^0+\text{_}x^5\cdot 2^1+\text{_}x^4\cdot 2^2+\text{_}x^3\cdot 2^3+\text{_}x^2\cdot 2^4+\text{_}x^1\cdot 2^5+\text{_}x^0\cdot 2^6\]

Answer 5

so i did it right?

Answer 6

I think so :O you erased it lolol so i don't remember XD

Answer 7

x6 + 6x5y + 15x4y2 +20x3y3 +15x2y4 +6xy5 + y6

Answer 8

The coefficient in front of each term is given by Pascal's Triangle.
Do you understand how to use Pascal's Triangle? :D

Answer 9

I get the coefficients part

Answer 10

but didn't I leave out the 2?

Answer 11

You have y's in place of your 2's.
It's probably easier to work with y's, but we need to put our 2's in and simplify.

Answer 12

ahh

Answer 13

so for "x^6" do I put a 2 after it

Answer 14

\[\Large\bf\sf x^6 + 6x^5\cdot2 + 15x^4\cdot2^2 +20x^3\cdot2^3 +15x^2\cdot2^4 +6x\cdot2^5 + 2^6\]

Answer 15

So that's what your expansion looks like with 2's instead of y's.
Now we need to multiply out the powers of 2, and then combine them with the coefficients in front.

Answer 16

So for the 3rd term,\[\Large\bf\sf 15x^4\cdot2^2\quad=\quad 15x^4\cdot4\quad=\quad 60x^4\]Understand what I did there? :o

Answer 17

yes

Answer 18

let's start from the beginnings

Answer 19

would it be 6x^30

Answer 20

Beginning of what? :x beginning with the first term?

Answer 21

ok im back

Answer 22

yes, first term

Answer 23

wait, i think its 12x^32... but that still sounds wrong

Answer 24

i finished the rest of it.. this is what i got: 12x32 + 60x4 + 120x3 + 120x2 + 70x

Answer 25

see how the exponents are going down by 1? I think the first exponent should be 5

Answer 26

are you there? @zepdrix

Answer 27

can you help me with the first term

Answer 28

x^6+6x^5⋅2

Answer 29

So the first one is fine right? x^6 = x^6.
Is that the one you were commenting on, or the next one?

Answer 30

^^ its up there

Answer 31

You wrote the first `two` terms.

Answer 32

Terms are separated by addition. :o

Answer 33

ohh.. they are separate... that makes more sense

Answer 34

sorry.. stupid moment there

Answer 35

ok so it will look like: x^6+12x^5 + 60x^4 + 120x^3 + 120x^2 + 70x

Answer 36

12x^5

Answer 37

\[\Large\bf\sf 20x^3\cdot2^3\quad=\quad 20\cdot2^3x^3\quad=\quad 20\cdot8x^3\]

Answer 38

Careful with your powers! Did you accidentally get 6 from 2^3?

Answer 39

woops..

Answer 40

oh let me fix the last 2...

Answer 41

wait it looks weird..

Answer 42

are 6x* 2^5 and 2^6 separate?

Answer 43

Yes.

Answer 44

but that doesn't make sense.... the exponents descend from 6 to 2... then they go " 5... 6" ?

Answer 45

x^6+12x^5 + 60x^4 + 160x^3 + 120x^2 + 12x^5 + 2^6

Answer 46

ok. im stupid. you simplify it to 64 right?

Answer 47

2^6 becomes 64

Answer 48

The last one? Yah 64 sounds right.

Answer 49

but what about 12x^5?

Answer 50

x^6+12x^5 + 60x^4 + 160x^3 + 120x^2 + 12x^5 + 64 <---- I mean, doesn't it look weird? the exponents go: 6, 5, 4, 3, 2, 5

Answer 51

when solving 6x⋅2^5 ... do you do 2^5 first?

Answer 52

It shouldn't be a coefficient of 12, and it should be to the 1st power.
Maybe we made a small typo somewhere.\[\Large\bf\sf x^6 + 6x^5\cdot2 + 15x^4\cdot2^2 +20x^3\cdot2^3 +15x^2\cdot2^4 +\color{orangered}{6x\cdot2^5} + 2^6\]The orange one there.

Answer 53

See how the 5th power is on the 2? Not the x.

Answer 54

so it should say 2x^5?

Answer 55

Oh .. yes :P we do the 2^5 first.. yer too fast for me lol

Answer 56

No it's not supposed to be x^5.. :o

Answer 57

wait whaatt then how is it supposed to be

Answer 58

The way it's written in orange.
And we need to simplify it from there.

Answer 59

but it was written like that already

Answer 60

Yes...

Answer 61

u said it was a typo...

Answer 62

It's not a typo.

Answer 63

so. 2^5 is 32.

Answer 64

so.... 192x?

Answer 65

finally! the exponents go: 6, 5, 4, 3, 2, 1, 0

Answer 66

yah looks good

Answer 67

can you help me with one more? its subtraction
(x – 4)^4

Answer 68

Did we miss any? Were you able to figure out the x^2 term ok?

Answer 69

yea

Answer 70

So you came up with something like this I think?\[\Large\bf\sf x^6+12x^5+160x^3+240x^2+192x+64\]

Answer 71

yes

Answer 72

wait u forgot a term

Answer 73

Oh I did haha XD

Answer 74

and where did u get the 240 from..

Answer 75

nvm

Answer 76

its x^6+12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64

Answer 77

Mmm ya looks good!

Answer 78

did you look at the one up there?

Answer 79

Ya the subtration makes this next one a little tricky.
Here is one way you can think of it.\[\Large\bf\sf (x-4)^4\quad=\quad (x+-4)^4\]So I rewrote it as addition, and the second term is negative.

Answer 80

here's the row for this one: 1 4 6 4 1

Answer 81

\[\large\bf\sf=\quad\text{_}x^4(-4)^0+\text{_}x^3(-4)^1+\text{_}x^2(-4)^2+\text{_}x^1(-4)^3+\text{_}x^0(-4)^4\]

Answer 82

i think that makes it more confusing

Answer 83

\[\large\bf\sf=\quad\color{royalblue}{1}x^4(-4)^0+\color{royalblue}{4}x^3(-4)^1+\color{royalblue}{6}x^2(-4)^2+\color{royalblue}{4}x^1(-4)^3+\color{royalblue}{1}x^0(-4)^4\]

Answer 84

ok I'm with u

Answer 85

Well the negative is very important, we can't just leave it as subtraction.
Because some of the powers are even, so the negatives will disappear.
And others are odd powers, so the negative will stick around.

Answer 86

ok

Answer 87

So we'll ignore the zero powers.\[\Large\bf\sf=\quad x^4+\color{royalblue}{4}x^3(-4)^1+\color{royalblue}{6}x^2(-4)^2+\color{royalblue}{4}x^1(-4)^3+(-4)^4\]

Answer 88

And then we just have a little bit of simplification.

Answer 89

Ugh site running slow D:

Answer 90

im using firefox now it works better on it

Answer 91

can we keep going

Answer 92

so far i got:

Answer 93

x^4 + (-16^3)

Answer 94

Ok looks good so far!

Answer 95

-4^2 is -16 right

Answer 96

\[\Large\bf\sf -4^2\quad=\quad -16\]\[\Large\bf\sf (-4)^2\quad=\quad 16\]Make sure you know the difference between these two!

Answer 97

The brackets let us know that we need to square the negative as well.

Answer 98

\[x^4 + (-16^3) + (96x^2)\]