partial fraction + binomial expansion

[sinful plank]

<@&791435371564892232>

[zenith patio]

2x+4/(x-1)(x+3)
A/x-1 + B/x+3

Multiply through by (x-1)(x+3) and equate to the numerator
A(x+3)+B(x-1)=2x+4
Ax+3A+Bx-B=2x+4
Compare
X(A+B)=x(2)
3A-B=4
Solve for a and b and sub back in

Or use substitution let x=1 to eliminate B
So u have 4A=6
A=3/2

And then x=-3

-4B=-2
B=1/2

[sinful plank]

yeah i got that already

i’m talking about part 2 rn

like how did the denominator become 2x(1-1/x)

i’m not sure what the q is trying to ask

[zenith patio]

They factored out x

[sinful plank]

why

[zenith patio]

because as x->infinity 1/x ->0

Wait uhhe

Idk 💀

[sinful plank]

wait can u explain what the question means first

[raw mango]

those are 2 different questions in entirety I believe

in both questions you solve for x

in 8i) multiply everything by -3 and then put the (1/3) and (1/9) inside brackets with x and x^2

then u got a quadratic

actually don’t do that because although its true it doesn’t progress anywhere

[raw mango]

but yeah multiply out the denominators and solve for x in a kinda complicated quartic function

[sinful plank]

why tho

i don’t understand what part 2 means

[raw mango]

@sinful plank oh I understand now

When x is large it basically means find the asymptote as x tends to infinity

What I would do is let y = 1/x then put the function in terms of y & then replace y back with 1/x

[sinful plank]

would be able to write out instead

so do i just substitute?

a bit confused

[raw mango]

You don’t HAVE to, but it may make it easier to visualise

[sinful plank]

still confused

why is this question phrased like that bruh

[raw mango]

@sinful plank ignore everything I said before, I was wrong

Its a binomial expansion question, some binomial expansions are only valid when x<1.

for part 2, as x is large, 1/x will be small, so subbing y = 1/x will allow the expansion to be valud

[sinful plank]

replace which value/constant with 1/x ?

after doing this then just expand as usual?

@raw mango

[raw mango]

yes

[raw mango]

It may be easier to substitute 1/x as a different variable, eg ‘y’ to follow the formula sheet, but this is your choice

[sinful plank]

but if i substitute x with 1/x here

[sinful plank]

i don’t get the same answer as the mark scheme

i just need to get the partial fraction correctly first