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# VERY HARD further maths question

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1. Let a,b and c be real numbers such that a+b+c=0 and let
(1+ax)(1+bx)(1+cx)=1+qx^2+rx^3
for all real x.Show that q=bc+ca++ab and r=abc
2. Expand out and equate coefficients.
3. (Original post by B_9710)
Expand out and equate coefficients.
Not going to work
4. (Original post by STRANGER2)
Not going to work
A shame, because it does
5. (Original post by STRANGER2)
Not going to work
Yes it does. And it's not even close to being a 'VERY HARD' question.
6. (Original post by STRANGER2)
Not going to work
(1+ax)(1+bx)(1+cx)=abcx^3+abx^2+ acx^2+bcx^2+ax+bx+cx+1=rx^3+qx^2 +1

(abc)x^3=rx^3 so abc=r
(ab+ac+bc)x^2=qx^2 so q=ab+ac+bc
(a+b+c)x=0x so a+b+c=0

7. Peasy.. As above.
If you couldn't do that..
8. I thought it was hard.
9. (Original post by The Financier)
(1+ax)(1+bx)(1+cx)=abcx^3+abx^2+ acx^2+bcx^2+ax+bx+cx+1=rx^3+qx^2 +1

(abc)x^3=rx^3 so abc=r
(ab+ac+bc)x^2=qx^2 so q=ab+ac+bc
(a+b+c)x=0x so a+b+c=0

Let's carry on with the question then: (I'm mad!)

Show that the coefficient of x^n in the series expansion (in ascending powers of x) of ln(1+qx^2+rx^3) is (-1)^n+1 Sn where
Sn= (a^n+b^n+c^n)/n
where n>1

Btw if it's getting too tough for anyone it's ok this is only for top maths students
(If u get this question right I'll carry on because question not finished)
10. (Original post by STRANGER2)
Let's carry on with the question then: (I'm mad!)

Show that the coefficient of x^n in the series expansion (in ascending powers of x) of ln(1+qx^2+rx^3) is (-1)^n+1 Sn where
Sn= (a^n+b^n+c^n)/n
where n>1

Btw if it's getting too tough for anyone it's ok this is only for top maths students
(If u get this question right I'll carry on because question not finished)
what have you tried?
11. Getting Zacken involved in this :P
12. (Original post by STRANGER2)
Let's carry on with the question then: (I'm mad!)

Show that the coefficient of x^n in the series expansion (in ascending powers of x) of ln(1+qx^2+rx^3) is (-1)^n+1 Sn where
Sn= (a^n+b^n+c^n)/n
where n>1

Btw if it's getting too tough for anyone it's ok this is only for top maths students
(If u get this question right I'll carry on because question not finished)
Well we know the quadratic can be factorised as above so you get a product of linear expressions inside the log function. Using log laws you can add these separately quite easily as you should remember the expansion of .
13. (Original post by The Financier)
(1+ax)(1+bx)(1+cx)=abcx^3+abx^2+ acx^2+bcx^2+ax+bx+cx+1=rx^3+qx^2 +1

(abc)x^3=rx^3 so abc=r
(ab+ac+bc)x^2=qx^2 so q=ab+ac+bc
(a+b+c)x=0x so a+b+c=0

I'm at GCSE level at the moment. I don't like the look of having to learn this... ;(
14. (Original post by Tasty Apple)
I'm at GCSE level at the moment. I don't like the look of having to learn this... ;(
It's not too bad really. You'll learn a simpler version at AS.
15. (Original post by STRANGER2)
Let's carry on with the question then: (I'm mad!)

Show that the coefficient of x^n in the series expansion (in ascending powers of x) of ln(1+qx^2+rx^3) is (-1)^n+1 Sn where
Sn= (a^n+b^n+c^n)/n
where n>1

Btw if it's getting too tough for anyone it's ok this is only for top maths students
(If u get this question right I'll carry on because question not finished)
I'm fairly sure you've just gotten this from a STEP question, in which case - work it out yourself instead of having people do it for you.
16. (Original post by B_9710)
Well we know the quadratic can be factorised as above so you get a product of linear expressions inside the log function. Using log laws you can add these separately quite easily as you should remember the expansion of .
Give me a worked solution or just quit and tell me you're not top maths student
if u do that I'll work it out for u
17. (Original post by morgan8002)
It's not too bad really. You'll learn a simpler version at AS.
You sure? Because there's a lot of letters there. And what does ^ mean?

Actually, don't tell me. I don't think I want to know...
18. (Original post by Tasty Apple)
You sure? Because there's a lot of letters there. And what does ^ mean?

Actually, don't tell me. I don't think I want to know...
It just means 'to the power of'
19. (Original post by Tasty Apple)
I'm at GCSE level at the moment. I don't like the look of having to learn this... ;(
Things often look difficult until you've studied them and practiced a fair amount. You might come across the first part in A-level maths and the second part in Further Maths.
20. (Original post by Tasty Apple)
You sure? Because there's a lot of letters there. And what does ^ mean?

Actually, don't tell me. I don't think I want to know...
It's just a shorthand for power because you can't write superscripts easily on a keyboard. x^n = .

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