Solving a Polynomial with Real Coefficients and Real Zeroes

[utkarshakash]

Homework Statement

Let f(x) = $x^{4}+ax^{3}+bx^{2}+cx+d$ be a polynomial with real coefficients and real zeroes. If |f(i)| = 1, (where $i = \sqrt{-1}$) then find a+b+c+d.

Homework Equations

The Attempt at a Solution

f(i) = 1-b+d+ci-ai
Taking modulus

|f(i)|= |1-b+d+i(c-a)|
$=(1-b+d)^{2}+(c-a)^{2}=1$

Simplifying
$a^{2}+b^{2}+c^{2}+d^{2}=2(b-d+bd+ac)$

But this takes me nowhere close to the answer. What else can I try?

 

[dcassell]

$=(1-b+d)^{2}+(c-a)^{2}=1$

This may or may not help... but the above equation looks very much like the equation for a unit circle centered around (1,0) where x=-b+d and y=c-a.

 

[utkarshakash]

This may or may not help... but the above equation looks very much like the equation for a unit circle centered around (1,0) where x=-b+d and y=c-a.

But this won't be of any help either.

 

[tiny-tim]

hi utkarshakash!

have you tried calling the roots (the zeroes) p q r and s ?

 

[gabbagabbahey]

Hmm... a little bit more background info might be helpful to suggesting an appropriate method of solving this for you. What course is this for? Are you allowed to use only algebraic methods (this thread is in the pre-calculus forum, so that is the assumption many posters might make), or can you use a little calculus as well?

The first idea that comes to mind for me is to try applying http://en.wikipedia.org/wiki/Sturm's_theorem]Sturm's[/PLAIN] theorem, to see under what restrictions on your coefficients, your polynomial will have all real roots (zeroes). But this method may or may not be appropriate to your course, and also may not work (I haven't tried it yet).

In any case, I would suggest the first thing you do is sketch out a few graphs of the possible forms your quartic can take if it is to have all real coefficients and roots (remember, the roots may not all be distinct, so you will need a few different graphs).

 

[utkarshakash]

Hmm... a little bit more background info might be helpful to suggesting an appropriate method of solving this for you. What course is this for? Are you allowed to use only algebraic methods (this thread is in the pre-calculus forum, so that is the assumption many posters might make), or can you use a little calculus as well?

The first idea that comes to mind for me is to try applying http://en.wikipedia.org/wiki/Sturm's_theorem]Sturm's[/PLAIN] theorem, to see under what restrictions on your coefficients, your polynomial will have all real roots (zeroes). But this method may or may not be appropriate to your course, and also may not work (I haven't tried it yet).

In any case, I would suggest the first thing you do is sketch out a few graphs of the possible forms your quartic can take if it is to have all real coefficients and roots (remember, the roots may not all be distinct, so you will need a few different graphs).

Hey I'm not supposed to take help of these advanced theorems like Sturm's Theorem. It is a question from the chapter Complex Numbers. Also I'm enrolled in a high school course so I have to use only algebraic methods.

 

[utkarshakash]

hi utkarshakash!

have you tried calling the roots (the zeroes) p q r and s ?

If I assume p,q,r,s to be the roots then I can find sum and products of the roots. What else?

 

[tiny-tim]

(the sum and product are a and d)

you can get the formulas for b and c also

 

[utkarshakash]

(the sum and product are a and d)

you can get the formulas for b and c also

You are wrong. The sum and products are -a and d and I have already found out the formulas for b and c. But simply adding them won't do me any good.

 

[gabbagabbahey]

Maybe consider a slightly more straightforward problem first: suppose you were given the polynomial $(x^2+\tilde{a}x+\tilde{b})(x^2+\tilde{c}x+\tilde{d})$, where the coefficients $\tilde{a}$, $\tilde{b}$, $\tilde{c}$ & $\tilde{d}$ are all real. Under what conditions would the polynomial have all real roots (zeroes)?