Year 12 Binomials

[ember kraken]

hi so idk where i'm going wrong here

markscheme

how is the coefficient of x^5 = -45/16

i thought it was -9/16

cuz that's what i got

[dusk bane]

ok so

you found the x^5 term of the

(3 - 1/2 x)^6 thing

right

thats what you just worked out

@ember kraken

[ember kraken]

yeah

and i got -9/16

[dusk bane]

but you want the x^5 term of the whole thing

[ember kraken]

huh

[dusk bane]

(5 + 8x^2)(3 - 1/2 x)^6

[ember kraken]

yeah

[dusk bane]

you want the x^5 term of that

so after you do stuff with the expanding on the ^6 bit

you need to times it by (5 + 8x^2)

right?

[ember kraken]

yeah

[dusk bane]

which will change it

[ember kraken]

oh i thought we also had to find the x^2 coefficient sub it into 8x^2

[dusk bane]

nop

ill give an example so like

so you said that the x^2 term is

-1215/4 x^2

[ember kraken]

something like this?

[dusk bane]

yeah so

we have -9/16 x^5

[ember kraken]

but what do i do with the x^2

[dusk bane]

so after timesing by (5 + 8x^2)

[ember kraken]

what abt x^2 cuz we need to get rid of that dont we

[dusk bane]

-9/16 x^5 (5 + 8x^2) will be

-9/16 * 5 * x^5 - 9/16 * 8 * x^2 * x^5

do you see how i expanded that

[ember kraken]

no what

[dusk bane]

the -9/16 x^5 term from your (3 - 1/2 x)^6 gets -9/16 x^5

and then after you times that by (5 + 8x^2)

[ember kraken]

i have no clue what u just did after that

[dusk bane]

lemme draw

expanding brackets

yeah?

@ember kraken

[ember kraken]

yeah i get u now

[dusk bane]

so do you see that

the red term will stay as x^5

all good

but the green term has x^5 times x^2 in it

[ember kraken]

yeah so is that gonna be x^7

[dusk bane]

exactly

so the red one is the only one that matters

BUT

theres another term we care about

so for every power of x you multiply with this bracket

the red term will keep it the same

the green term will basically

[ember kraken]

what

[dusk bane]

add 2 to the power

right?

[ember kraken]

so do we ignore the green

[dusk bane]

well no because

theres a power of x where

the green one turns into a x^5

[ember kraken]

where

when

so what do i do

[dusk bane]

well have a think

the green term is times by 8x^2

aka adding 2 to the power of x

[ember kraken]

i dont know

[dusk bane]

and you wanna end up with x^5

so what do you times it by

[ember kraken]

-2

[dusk bane]

what power of x would multiply with x^2 to give x^5

[ember kraken]

oh

3

[dusk bane]

mmmm

so if you multiplied this by the x^3 term

[ember kraken]

i thought u meant to go from x^7 to x^5

huh

[dusk bane]

the green one would be x^5 now right?

[ember kraken]

i swear it would be 3??

yes..?

[dusk bane]

yeah x^3 times x^2 is x^5

[ember kraken]

yeah

[dusk bane]

so if you take an x^3 term

and times it by (5 + 8x^2)

[ember kraken]

wheres this x^3 and x^2 stuff all coming from 😭

[dusk bane]

the 8x^2 times it gives you x^5

x^2 is from 8x^2

[ember kraken]

why so many things to do for 5 marks

[dusk bane]

lets quickly just recap

cuz i think we are a bit lostr

[ember kraken]

girl i am lost 😭

[dusk bane]

so you have already found the x^5 term from (3 - 1/2 x)^6 right

[ember kraken]

only bit i know is the expanding of -9/16 (5+8x^2)

[dusk bane]

its -9/16 x^5

and when we multiplied this by (5 + 8x^2)

we found that the times by 5

[ember kraken]

yeah that

[dusk bane]

keeps it as an x^5 thing right

[ember kraken]

whoops

yeah

[dusk bane]

awesome

but you also said

if you multiplied this (5 + 8x^2) thing by an x^3

[ember kraken]

ik that green thing

[dusk bane]

then the 8x^2 times the x^3 would give you another x^5

[ember kraken]

idk what to after the green thing and why

[dusk bane]

right?

[ember kraken]

yeah

[dusk bane]

but the (3 - 1/2 x)^6

will have an x^3 term in it

[ember kraken]

yeah

[dusk bane]

so thatll be the thing you times by the 8x^2

to give you an x^5

[ember kraken]

why

[dusk bane]

because x^3 times x^2 is x^5

x^3 is in the x^3 term of (3 - 1/2 x)^6 because its just what it is

x^2 is in 8x^2

[ember kraken]

yeah so what do i do

[dusk bane]

from here

[ember kraken]

how come we're ignoring the x^7 bit

[dusk bane]

read and infer

because we only want the x^5 term

[ember kraken]

is there any other way of working around it because im still not understanding it

[dusk bane]

x^7 isnt what we are after

im gonna find the first few terms of this thing hold on

[ember kraken]

im still not getting it

i still do not understand any of it

like

nothings going in

[dusk bane]

actually what im gonna do is

[ember kraken]

and i fumbled my maths mock and i need to reteach myself AS level maths again for my resit in september

[dusk bane]

lets pretend that

instead of (3 - 1/2 x)^6 because its gross

[ember kraken]

and i need to teach myself A2 aswell cuz ik we're not gonna have enough time to fully complete A2

[dusk bane]

im just gonna use (1 + x)^6

so the numbers are nicer

idea will be the same tho

[ember kraken]

but i've already got the terms from (3-0.5x)^6

[dusk bane]

yeah but not all simplified

which is the issue

so im gonna do a slightly different but basically the same question

which is

find the x^5 term for (5 + 8x^2)(1+x)^6

[ember kraken]

do i basically do the term for x^2 multiplied by x^3

for the green bit

[dusk bane]

well the x^3 has stuff with it

which you need to find

[ember kraken]

what stuff

oh my god 😭

[dusk bane]

that is why im doing this adapted simpler one

so i dont have to deal with the shitty coefficients

so

x^5 term in (5 + 8x^2)(1 + x)^6

and im just gonna give you for free that

(1 + x)^6 is:

x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1

ok?

[ember kraken]

what

yeah

[dusk bane]

im giving you that for free

its just binomial

to save time working it out

[ember kraken]

so 6x^5(5+8x^2)

?

[dusk bane]

well so thats part of it

what happens if you do that

[ember kraken]

thats 30x^5 + 48x^7

[dusk bane]

ok nice

[ember kraken]

thats what we get

[dusk bane]

now im gonna pick a different term

like eg the 15x^4 bit

15 x^4 (5 + 8x^2)

thats gonna be what like

75x^4 + 120 x^6

smth like that

notice how theres the first bit

which is the same power as the thing you timesed by originally

like for yours it was an x^5

mine its an x^4

yeap?

cuz times by 5 doesnt change the power of x

@ember kraken

[ember kraken]

hang on let me read this

[dusk bane]

yep calm

[ember kraken]

yeah

[dusk bane]

the term after it the power is increased by 2, for yours its the x^7 and for mine its the x^6

[ember kraken]

yeah

i get that bit

[dusk bane]

so i will say that

[dusk bane]

there is a term in this

where the second thing after multiplying by (5 + 8x^2)

is an x^5

which one

[ember kraken]

x^3

[dusk bane]

yeah its the 20x^3

if you do 20x^3 (5 + 8x^2) youll get

100x^3 + 160x^5

[ember kraken]

yeah

[dusk bane]

so in our whole expansion

[dusk bane]

we will get the 30x^5 from your thing here

with the 6x^5 thing

[dusk bane]

but also the 160x^5 from here

using the 20x^3 term

does that make sense?

so all together we will have 160x^5 AND 30x^5

[ember kraken]

what

im confused now

like

really confused

like we're back to square 1 confused

[dusk bane]

so you said that

if you multiplied that whole

x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1 mess

by the (5 + 8x^2)

if you use the 6x^5 term youll get a 30x^5

[ember kraken]

did i say that wtf

thats a bit goofy

[dusk bane]

because 6x^5 (5 + 8x^2 ) is 30x^5 + 48x^7

[ember kraken]

yeah

[dusk bane]

theres the 30x^5 in there

[ember kraken]

i dont get the 48x^7 bit

[dusk bane]

well the x^7 isnt important

because we are only after the x^5 thing

[ember kraken]

so why cant we just ignore it

[dusk bane]

the 48x^7 is stuff we get from multiplying out the bracket

well i am ignoring it

but im explaining where the 30x^5 comes from

it comes from expanding that

and then we throw away the 48x^7 because its not important

[ember kraken]

yeah so from the exam question cant we just ignore -9/2x^7

[dusk bane]

yeah

anyway

but we also said that

[ember kraken]

so we can just ignore that

how come we went on a ramble about it then 😭

[dusk bane]

if you times the (5 + 8x^2) by the 20x^3 term

the ramble was about how to find the 30x^5

but if i just said 6x^5 (5 + 8x^2) gives 30x^5 youd have got confused again

[ember kraken]

doesnt it just come from 6x^5 * 5

[dusk bane]

yeah

which you have beacuse

you are multiplying by (5 + 8x^2)

anyway

[ember kraken]

yeah but

[dusk bane]

we also said that if you use the 20x^3

[ember kraken]

isnt it -9/16 and not 6

for the coefficient of x^5

[dusk bane]

yeah but youll see

cuz you havent actually worked out the term for this next bit in your question

and im too lazy to find it myself

so i used a different bracket

[ember kraken]

whats a term for the next bit

[dusk bane]

this is what im trying to explain

as i said

we also said if we multiply the (5 + 8x^2) by the 20x^3 term

you get 100x^3 + 160x^5

as you can see

another x^5 term right

[ember kraken]

yeah but

why

[dusk bane]

because x^3 times x^2 is x^5

[ember kraken]

how come we cant just ignore the -9/2x^7

[dusk bane]

we can

which is what i said

[ember kraken]

u said we could ignore the other term

but how come we're not in this example

[dusk bane]

cuz im explaining where everything comes from

[ember kraken]

i know but

icl i would've just ignored it

[dusk bane]

if i just said we get 160x^5 from the x^3 term and 30x^5 from the x^5 term

youd have no idea what the fuck i mean

[ember kraken]

if this came up on my mock i would've ignored the x^7 bit

[dusk bane]

are you happy with that

with the 20x^3 term

we get 160x^5

and with the 6x^5 term we get 30x^5

[ember kraken]

no

[dusk bane]

why not

[ember kraken]

im sorry but where did 20 and 160 bits come from

[dusk bane]

20x^3 from here

[ember kraken]

are we using the (3-0/5x)^6 terms

[dusk bane]

no

[ember kraken]

but thats from the example though

thats from a different example

[dusk bane]

yeah but you havent done the working for the bit im trying to explain

[ember kraken]

im so confused

[dusk bane]

we are currently doing the question

find the x^5 term from (5 + 8x^2)(1+x)^6

im giving the freebie that

(1+x)^6 is

x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1

cool?

[ember kraken]

6x^5

[dusk bane]

yep so

6x^5 times the (5 + 8x^2)

gives you 30x^5

[ember kraken]

30x^5 and

[dusk bane]

and a thing we dont care about

[ember kraken]

48x^7

yeah

[dusk bane]

ok

but youll see that

[ember kraken]

so we completely discard 48x^7?

[dusk bane]

yep

now try do the same expansion with 20x^3

[ember kraken]

and we also discard the -9/2x^7 in the question?

why

[dusk bane]

youll see

[ember kraken]

why do we expand

wheres 20x^3 come from

[dusk bane]

from the middle of x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1

[ember kraken]

160x^5

[dusk bane]

mmm

another x^5 term right

[ember kraken]

vut why do we multiply 20x^3

teag

yeah

[dusk bane]

because the x^2 from the 8x^2 with the x^3 from the 20x^3

gives you an x^5

and the question wants us to find x^5 terms

right?

we use 20x^3 because it gives us an x^5 term which we want

all the other terms dont give us x^5

[ember kraken]

yeah i get that bit now

[dusk bane]

ok so

[ember kraken]

what do i do now

[dusk bane]

in total we will have 30x^5 from the 6x^5

and 160x^5 from the 20x^3

[ember kraken]

i've never seen a question like this before and this question is on a past paper

yeah

[dusk bane]

so combined

how many x^5

[ember kraken]

3

[dusk bane]

nope

[ember kraken]

PH

4

my bad

[dusk bane]

you have 160x^5 and 30x^5

[ember kraken]

yeah

[dusk bane]

so together its

190x^5

[ember kraken]

and we have 30x^5 and 6x^5

no wait

[dusk bane]

nah cuz the 6x^5 is from before exapdning

[ember kraken]

6x^5 and 20x^5

[dusk bane]

we want after expanding

[ember kraken]

oh

si uts hyst 2

so its just 2

[dusk bane]

yeah

[ember kraken]

so the answers 2?

got it

[dusk bane]

but they give you 190x^5

[ember kraken]

what

oh

we add now?

so 190x^5

[dusk bane]

yeah youd add all the terms

ok so

did that kinda make sense

[ember kraken]

nope but

nope im just doing what ur doing

[dusk bane]

cuz your question is the exact same with worse numbers

fair enough we will just do your question anyway

so (5 + 8x^2)(3 - 1/2 x)^6

find the x^5 term

right

[ember kraken]

yeah

[dusk bane]

ok so you did part of it

[ember kraken]

-45/16 x^5 - 9/2x^7

what do i do from there

[dusk bane]

thats from the -9/16 x^5 out of the (3 - 1/2 x)^6 right

so similar to our last question

[ember kraken]

yeah

[dusk bane]

we will also need the x^3 term from this

(3 - 1/2 x)^6

[ember kraken]

which is -45/2 x^3

[dusk bane]

ok so

multiply that by your (5 + 8x^2)

what do you find

[ember kraken]

-45/2x^3(5+8x^2)

ummm

give me like

5 minutes

[dusk bane]

yeah nws

[ember kraken]

so 225/2 x^3

-180 x^5

[dusk bane]

im gonna quickly

so according to online calculator its -135/2 x^3

but thatll give you

[ember kraken]

i got -180

what

oh the x^3 term

[dusk bane]

as in

yeah

[ember kraken]

hang on let me redo it again

[dusk bane]

not -45/2 x^3

[ember kraken]

yeah i got it now

[dusk bane]

ok cool

then if you do the (5 + 8x^2) times that

you get this

just to save you working

[ember kraken]

yeah

[dusk bane]

those should be negative actually but yeah

so you see again

x^5 term

[ember kraken]

wait wait wait

[dusk bane]

should be negative cuz its -135/2

i typed in 135/2 cuz i was lazy

[ember kraken]

-135/2 * 5 shouldnt that be -675/2

then also be - 540

[dusk bane]

yeah both should be negative

[ember kraken]

then we just combine negative 540 and -45/16

[dusk bane]

exactly

both the x^5 terms

[ember kraken]

ohh right

i get it now

do u know where i can find questions like this

[dusk bane]

errrr ill have a quick search

[ember kraken]

cuz the ones in the book only ask for finding the coefficients

[dusk bane]

but the idea is you need to

find which terms give the right power

when you times by your extra bracket

[ember kraken]

huh

[dusk bane]

https://studywell.com/wp-content/uploads/2021/03/BinomialExpansionExamQuestions.pdf some of these?

[ember kraken]

ye i've just printed it off now

for this type of question

do u only need to know what method to carry out

and not the why bits

i got an E in my AS-level maths mock

could've been a D instead

idk how i fumbled it so badly though

cuz my tests throughout the year was fine

[dusk bane]

cant find many others

[ember kraken]

i was getting decent passes

i just printed some off pmt too

[dusk bane]

well i mean

if you dont know the why and you forget

[ember kraken]

do u reckon i can get a B in the end

[dusk bane]

you are completely fucked

[ember kraken]

oh

[dusk bane]

yeah i reckon you can get a B

[ember kraken]

cuz i need a B

[dusk bane]

just practice

[ember kraken]

idk how the logic works though

i got an A in my physics mock

and an E in my maths

idk how that works or correlates

[dusk bane]

idk either

[jade root]

All this for a binomial question rip

[ember kraken]

girl idk either

im trying rn to make sure i understand everything in AS

then ill do past paper at home

then ill see if im capable of moving onto A2

and i need to go over stats and mechanics too

cuz i got 15/60 on that

28/100 on pure

[jovial cove]

fr

[zinc geyser]

@ember kraken see if this helps