Home » Ask & Discuss » Mathematics. » Algebra « Back to Discussion

Algebra

Forum Expert

Joined:	28 Feb 2007
Post:	2173

19 Jul 2008 11:25:32 IST

0 People liked this

504

Yet another polynomial prob

None

f(z) is a non-constant polynomial with real coefficients such that for some real number a, we have

$f(a) \ne 0, f$

Prove that f(z) cannot have all roots real.

I happen to know the official solution. So, only original looking solutions get rated.

Share this article on:

Comments (2)

abhishek sinha

Forum Expert

Joined: 18 Dec 2007

Posts: 926

19 Jul 2008 18:09:12 IST

Like 3 people liked this

write $f(x)=c\prod(x-\alpha_{i})}$ ,where $\alpha_{i}$ s are the roots of the polynomial

given that f(a) !=0 whch means none of the roots are a .

Now we get f'(a)=0

so taking logarithm of both sides and then taking derivative at x=a , we obtain

$\sum{\frac{1}{a-\alpha_{i}}=0$ ......(2)

again taking the derivative of f'(x ) at x=a and imposing the condition that f''(a)=0 , we get using (2)

$\sum{\frac{1}{(a-\alpha_{i})^2}=0$ ....(3)

but it is given that a is real . So if all the roots are real then surely (3)can't hold true .

Hence proved .

Hari Shankar

Forum Expert

Joined: 28 Feb 2007

Posts: 2173

20 Jul 2008 07:37:12 IST

Like 0 people liked this

thats the solution i was looking for!

A bit of clarification may be required here:

We have for a polynomial f(x) with roots $\alpha_i$ :

$\frac{f$

Since f'(a) = f"(a) = 0, we have

$\sum_{i=1}^n \frac{1}{(x-\alpha_i)^2} = 0$

Quick Reply

Reply

	Some HTML allowed.
	Keep your comments above the belt or risk having them deleted.
	Signup for a avatar to have your pictures show up by your comment
	If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team

Free Sign Up!

Preparing for IIT-JEE ?

Arihant Revision Package for IIT JEE - Books, Practice Tests + Rank Predictor

@ INR 1,995/-

For Quick Info

Find Posts by Topics

Physics.

Topics

9391

20087

3671

2186

7614

2826

2636

Mathematics.

Chemistry.

Biology

Institutes

Parents

Board

Fun Zone