Roots of equations

TheCenturion · 22 Apr 2011 at 21:20

Harro everyone

The textbook doesn't help. We haven't covered it in class. No one is online on facebook that can help.

OcUK is my last resort

The equation z^3 + 4z^2 - 3z + 1 = 0 has roots a, b and c

Find the values of:

a + b + c
ab + bc + ac
abc

Somehow I need to factorise the equation, cos then I'll get values for -a, -b and -c. But how? This question comes up in every exam paper (further pure 1) and I've got no idea how to do it

Thanks friends

muon · 22 Apr 2011 at 21:27

You sure, that's the question?

It doesn't have simple solutions.

http://www.wolframalpha.com/input/?i=z^3+++4z^2+-+3z+++1+=+0

In FP1 I suspect all they want is you to use the factor theorem well.

http://en.wikipedia.org/wiki/Factor_theorem

marc_howarth · 22 Apr 2011 at 21:28

That's a horrible question.

Usually there's an obvious root say -1 or 1 which you can use.

When there's an obvious root to be found, say a. You can divide the cubic by (x - a) and then factorise the remaining quadratic.

TheCenturion · 22 Apr 2011 at 21:29

I've gotten a + b + c

That's - 4/1 = -4

Not sure how to proceed from there

TheCenturion · 22 Apr 2011 at 21:31

marc_howarth said:
That's a horrible question.

It's a standard FP1 question I'm afraid. There's specific rules for identities that you have to use. But in the text book, these rules are only explained for quadratics, and not cubics :rolleyes:

Still using the rule, I got the answer to a + b + c, now need to try to get the others.

marc_howarth · 22 Apr 2011 at 21:33

It's a horrible question in terms of having to factorise it. If there are identities that you can use to avoid factorising I guess it's not too bad. Been a while since I did FP1 though

Six7th · 22 Apr 2011 at 21:34

Does seem pretty nasty. Usually you'd just put a number in for z until it equals zero, then you know it is a root and the factor is (x minus whatever the number is). Using that you can then create a quadratic factor, and factorise that to get the other two.

Edit: beaten..and realised it's FP1 not C1

TheCenturion · 22 Apr 2011 at 21:37

I think I've found the rules. No thanks to the textbook or my teacher :rolleyes:

a+b+c = -b/a
ab+bc+ac = c/a
abc = -d/a

For the equation ax^3 + bx^2 + cx + d

so a + b + c = -4/1 = -4
ab + bc + ac = -3/1 = -3
abc = -1/1 = -1

Them's the answers apparently. Still, I'm thinking wtf

marc_howarth · 22 Apr 2011 at 21:47

It's confusing in the way that you are using a, b, c as the roots of the equations as well as the coeffecients of each term.

think of:
ax^3 + bx^2 + cx + d = 0 having roots f, g and h:
then
f + g + h = -b/a
fg + gh + fh = c/a
fgh = -d/a

Six7th · 22 Apr 2011 at 21:47

If there's nothing in your textbook then bottom of page 8 and top of 9 of this may do some explaining, http://www.mathshelper.co.uk/OCR FP1 Revision Sheet.pdf

TheCenturion · 22 Apr 2011 at 21:49

marc_howarth said:
It's confusing in the way that you are using a, b, c as the roots of the equations as well as the coeffecients of each term.

think of:
ax^3 + bx^2 + cx + d = 0 having roots f, g and h:
then
f + g + h = -b/a
fg + gh + fh = c/a
fgh = -d/a

Yeah that's true. It's only because it was alpha beta gamma initially, but that's very long winded to write.

Six7th said:
If there's nothing in your textbook then bottom of page 8 and top of 9 of this may do some explaining, http://www.mathshelper.co.uk/OCR FP1 Revision Sheet.pdf

Wow thanks mate, they've listed all the identities there!

muon · 22 Apr 2011 at 21:50

When did they start doing rule based mathematics at A-level? How is that pure maths?

I hope they teach you to prove those rules.

Six7th · 22 Apr 2011 at 21:58

ghost101 said:
When did they start doing rule based mathematics at A-level? How is that pure maths?

I hope they teach you to prove those rules.

Unfortunately on me exam board (OCR), there's seems to be very little of deriving things from first principles. They teach you in the book but it's never needed in the exam.

TheCenturion · 22 Apr 2011 at 22:12

I hate fp1. It's a pile of ****

I need to get an A but...

bhavv · 22 Apr 2011 at 22:20

Maths threads need to be BANNED from these forums for making me feel like a stupid

Aod · 22 Apr 2011 at 22:34

bhavv said:
Maths threads need to be BANNED from these forums for making me feel like a stupid

agreed!

Philtor · 22 Apr 2011 at 23:01

TheCenturion said:
Harro everyone

The textbook doesn't help. We haven't covered it in class. No one is online on facebook that can help.

OcUK is my last resort

The equation z^3 + 4z^2 - 3z + 1 = 0 has roots a, b and c

Find the values of:

a + b + c
ab + bc + ac
abc

Somehow I need to factorise the equation, cos then I'll get values for -a, -b and -c. But how? This question comes up in every exam paper (further pure 1) and I've got no idea how to do it

Thanks friends

You've put me back on the drink again.

BetaNumeric · 23 Apr 2011 at 00:04

ghost101 said:
When did they start doing rule based mathematics at A-level? How is that pure maths?

It's about algebraic identities, how is it not pure mathematics, in the A Level sense?

ghost101 said:
I hope they teach you to prove those rules.

It's trivial to prove. In fact you can prove the general result in precisely the same manner but due to lack of LaTeX support I'll do the cubic case.

f(x) = ax^3 + bx^2 + cx + d. Let the three roots be A, B, C, ie f(A) = f(B) = f(C) = 0. By factorisation theorems if f(A) = 0 then f(x) is exactly divisible by x-A and likewise for B and C. Thus f(x) is exactly divisible by (x-A)(x-B)(x-C). The coefficient of x^3 in that is 1 so we overall scale by a to get f(x) = a(x-A)(x-B)(x-C). This has all been done without doing actual expansions and factorisations. We expand this form of f(x) and equate the two formulae, f(x) = a(x-A)(x-B)(x-C) = ax^3 + a(A+B+C)x^2 + a(AB+BC+CA)x + a(ABC) = ax^3 + bx^2 + cx + d. Therefore b = a(A+B+C), c = a(AB+BC+CA), d = a(ABC). QED.

The generalise result for order N polynomials is done in a likewise manner.

Six7th said:
Unfortunately on me exam board (OCR), there's seems to be very little of deriving things from first principles. They teach you in the book but it's never needed in the exam.

Nice signature, though it isn't gauge invariant

muon · 23 Apr 2011 at 01:14

BetaNumeric said:
It's about algebraic identities, how is it not pure mathematics, in the A Level sense?

But simply using the identities given in the formula book shows no understanding? It's not even applying the factor theorem. At A-level you begin to understand everything from first principles.

It's trivial to prove. In fact you can prove the general result in precisely the same manner but due to lack of LaTeX support I'll do the cubic case.

f(x) = ax^3 + bx^2 + cx + d. Let the three roots be A, B, C, ie f(A) = f(B) = f(C) = 0. By factorisation theorems if f(A) = 0 then f(x) is exactly divisible by x-A and likewise for B and C. Thus f(x) is exactly divisible by (x-A)(x-B)(x-C). The coefficient of x^3 in that is 1 so we overall scale by a to get f(x) = a(x-A)(x-B)(x-C). This has all been done without doing actual expansions and factorisations. We expand this form of f(x) and equate the two formulae, f(x) = a(x-A)(x-B)(x-C) = ax^3 + a(A+B+C)x^2 + a(AB+BC+CA)x + a(ABC) = ax^3 + bx^2 + cx + d. Therefore b = a(A+B+C), c = a(AB+BC+CA), d = a(ABC). QED.

The generalise result for order N polynomials is done in a likewise manner.

Thanks. That is a question which requires understanding.

Philtor · 23 Apr 2011 at 01:17

ghost101 said:
But simply using the identities given in the formula book shows no understanding? At A-level you begin to understand everything from first principles.

Thanks. That is a question which requires understanding.

No s***

Surely the bloody computer can work it out? lol !