# Another proof

1. Sep 18, 2009

### nietzsche

Hi everyone. I know i keep posting all these questions, but each question in my textbook just keeps on bringing on new challenges. Is there a rule against posting to many questions?

1. The problem statement, all variables and given/known data

$$\text{Prove that if} \begin{math} x \end{math} \text{and} \begin{math} y \end{math} \text{are not both 0, then} \begin{equation*} x^4+x^3y+x^2y^2+xy^3+y^4>0 \end{equation*}$$

2. Relevant equations

N/A

3. The attempt at a solution

This is the second part of a question I posted earlier (https://www.physicsforums.com/showthread.php?t=338240). I'm guessing the questions are related somehow, but all the methods used on the earlier question don't seem to work on this question.

I tried grouping all the positive terms ($$x^4, y^4, x^2y^2$$) and separating the equation based on those, and it works for the cases where x and y are both positive or both negative, but when they have opposite signs, it's impossible to figure out.

Any hints? I know that multiplying by (x-y) gives $$x^5-y^5$$, but I don't know how much that helps.

Last edited: Sep 18, 2009
2. Sep 18, 2009

### Elucidus

Iis the last term in the inequality supposed to be y2 or y4?

--Elucidus

3. Sep 18, 2009

### nietzsche

ah yes, thank you, it's been fixed

4. Sep 18, 2009

### emyt

it's just the same as the previous part isn't it?

5. Sep 18, 2009

### nietzsche

well, i've tried this:

\begin{align*} x^4+x^3y+x^2y^2+xy^3+y^4 &> 0\\ x^4+y^4+xy(x^2+xy+y^2) &> 0 \end{align*}

and we know from the previous part that x^2 + xy + y^2 is positive, but there's that xy term which doesn't make it as clear cut as the last question.

6. Sep 18, 2009

### emyt

(x^5 - y^5) / (x-y) ?

btw, check your inbox if you haven't

7. Sep 18, 2009

### Elucidus

Consider cases where x > y and x < y.

(The case where x = y is more easily dealt with in the original polynomial form.)

--Elucidus

8. Sep 18, 2009

### emyt

yes, that's why I was confused - he came up with the same proof in his last thread

9. Sep 18, 2009

### nietzsche

haha, i guess come up with that in my last thread. oops.