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Will be visible here after 2015!Wow. You're motivated.
Thanks.
Re: The Further Maths Thread
Assalam-o-Alaikum...!
Hamid:.
"If the roots of the equation x^4 - px^3 + qx^2 - pq x + 1 = 0 are α, ß, Γ and delta(D) show that:
( α + ß + Γ) ( α + ß + D) ( α + Γ + D) ( ß + Γ + D) = 1"
You can observe that, in the product "( α + ß + Γ) ( α + ß + D) ( α + Γ + D) ( ß + Γ + D)," each of the expression consisting of the sum of the roots of the equation "x^4 - px^3 + qx^2 - pq x + 1 = 0" forms a pattern.
Let's represent the sum "α + ß + Γ + D" by "S" to try to simplify the hairy product.
Then, ( α + ß + Γ) ( α + ß + D) ( α + Γ + D) ( ß + Γ + D) = ( S - D ) ( S - Γ ) ( S - ß ) ( S - α )
= S^4 - DS^3 - ΓS^3 + ΓDS^2 - ßS^3 + ßDS^2 + ßΓS^2 - ßΓDS - αS^3
+ αDS^2 + αΓS^2 - αΓDS + αßS^2 - αßDS - αßΓS + αßΓD
= S^4 - ( α + ß + Γ + D )*S^3 + ( αß + ßΓ + ΓD + Dα + αΓ + ßD )*S^2
- ( αßΓ + ßΓD + ΓDα + αßD)*S + αßΓD
= S^4 - (∑ α)*S^3 + (∑ αß)*S^2 - (∑ αßΓ)*S + αßΓD
Now, from the equation "x^4 - px^3 + qx^2 - pq x + 1 = 0," ∑ α = p = S; ∑ αß = q; ∑ αßΓ = pq; αßΓD = 1
This leads to, ( α + ß + Γ) ( α + ß + D) ( α + Γ + D) ( ß + Γ + D) = p^4 - (p)*p^3 + (q)*p^2 - (pq)*p + 1
= p^4 - p^4 + q*p^2 - q*p^2 + 1
= 1
abcd:
As regards the method used to find sums of powers of roots of polynomial equations, it's quite simple but requires a clever method. I'll demonstrate the method using a quadratic equation.
Consider the quadratic equation x^2 - 4x + 5 = 0 with roots α and β.
If you multiply the cubic equation with x^n, you get
x^(n+2)-4x^(n+1)+5x^n=0
Since both α and β satisfy this equation,
α^(n+2)-4α^(n+1)+5α^n=0 and
β^(n+2)-4β^(n+1)+5β^n=0
Adding these two equations, we have
∑α^(n+2) - 4 ∑α^(n+1) + 5∑α^n = 0
If you denote ∑α^n by S
S(n+2) - 4 S(n+1) + 5 S
We know that α + β = 4 and αβ=5.
So α^2 + β^2 = (α + β)^2 - 2αβ = 4^2 - 2*5 = 6
Putting n = 1, S(3) - 4 S(2) + 5 S(1)=0
We know that S(1) = 4 and S(2) = 6, so S(3) = 4*6 - 5*4=4
I guess you understand the method and can apply it to equations of higher order.
Hope that helps.....
Hey can you help me .... i am unware from which books i should study Further Mathematics? wheni saw the past papers it was out of everything given in P1 & P3 booksRe: The Further Maths Thread
Oops...;p I guess I had allocated a misleading notation for the sum "α + ß + Γ + D." I should've used the symbol "S(1)" instead of just "S" to represent the sum, to avoid any type of confusion.
So, wherever I wrote "S," I indeed meant S(1). Accordingly, the product "( α + ß + Γ) ( α + ß + D) ( α + Γ + D) ( ß + Γ + D)" in the question lead to the expression "[S(1)]^4 - (∑ α)*[S(1)]^3 + (∑ αß)*[S(1)]^2 - (∑ αßΓ)*S(1) + αßΓD" upon simplification.
Then, from the equation "x^4 - px^3 + qx^2 - pq x + 1 = 0," ∑ α = p = S(1); ∑ αß = q; ∑ αßΓ = pq; αßΓD = 1
Leading to, ( α + ß + Γ) ( α + ß + D) ( α + Γ + D) ( ß + Γ + D) = [p]^4 - (p)*[p]^3 + (q)*[p]^2 - (pq)*p + 1
= p^4 - p^4 + q*p^2 - q*p^2 + 1
= 1
Note that none of S(2), S(3) or S(4) is involved in any of the expressions I wrote.
Hopin' I removed your confusion now....
P.S.
To be honest, I don't know of any further maths book covering this "elegant & exceptionally-simple + shortcut" method. However, the past-papers are full of questions on it. Just have a look at some of the questions about polynomials there (along with the corresponding er's/ms's), and you'll master this method very soon....!
Hey there! As long as you do Further Maths after AS Maths you should be fine for almost every topic. Make sure you have a solid background on statistics (ie. go through the S1 and S2 textbooks). Mechanics can be done almost completely without previous knowledge. The reason for doing it after AS maths is that you do need a background on vectors, differentiation and integration.
Listen, I am doing my AS level now and will finish it next month. I had taken Additional Maths in IGCSE and had got 97% so should I take Further Maths in A2?
I am doing it without any tutor
Interest is what you need. This is very less now a days specially for Math. I D K why most people chose bio thinking math is hard.Listen, I am doing my AS level now and will finish it next month. I had taken Additional Maths in IGCSE and had got 97% so should I take Further Maths in A2?
In A2, I have Accounts (A level), Eco (A level), Maths (A level), Physics (AS level), English (AS level). So do i take Further Maths next year?
I am very much interested in Maths... Ua suggestion please...?
Just kidding (my attitude ) GO FOR IT...
What if we fail?Get this http://www.amazon.com/Further-Pure-Mathematics-Brian-Gaulter/dp/0199147353
i used it but beware using these books urself is the recipe for disaster according to me ull get bogged down immediately as the explanation of examples like all maths books takes a lot of time and skill to understand so it wud be better if u get some teacher for it .
For mechanics i have no clue i had teachers and they just did all the theory and concepts and then we straightaway dived into past papers this technique worked really well in the end.
And to all those scaring the hell out of people giving or hoping to give FM eat up all your words because if the person giving FM does proper practice he can easily get an A* i did it myself so anyone feeling scared that theyll ruin their grades get motivated and start taking FM seriously it aint easy but if u can get the grade ure going to be popular in town
Okay... Thank you so much..Interest is what you need. This is very less now a days specially for Math. I D K why most people chose bio thinking math is hard.
Well, talking on your topic, you are fool to ask advice with this good qualifications in you.
Dont lose this interest. Its very challenging.
Have a look into past papers.
Depends on your SMART WORK brother, but looking at your performance and your zeal, I think you will ACE it.Okay... Thank you so much..
And will it be too hectic for me to study 6 subjects in the next year (depending on FM)
I mean how many hours we need to practice to be perfect in FM??
QUICK ASYMPTOTES GUIDE
Okay bro... Thnxxxx a ton...!!Depends on your SMART WORK brother, but looking at your performance and your zeal, I think you will ACE it.
Practice is must man.
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