Proving that a nth degree polynomial is > (or < etc.) some constant.

[QuarkCharmer]

Homework Statement

Let $x\in$R,
Prove that if x>2 then $x^4 - 8x^3+24x^2-32x+16$

Homework Equations

The Attempt at a Solution

So far I have only learned proofs involving even and odd numbers, that sort of thing. I'm not really sure how to approach this one. I was thinking that a proof by cases would suffice, so:

Proof:
Case I: Assume that x>2, and let S = (0,$\infty$). If follows that,
x = (m+2) for some m$\in$S.

And so,
$x^4 - 8x^3+24x^2-32x+16 = (m+2)^4 - 8(m+2)^3+24(m+2)^2-32(m+2)+16$

but now I see the problem becoming too difficult. We haven't learned about epsilon-delta et al. yet. I can see that the conclusion is ALWAYS true, so really the proof is trivial, but I am not sure how to say that.

 

[Curious3141]

Homework Statement

Let $x\in$R,
Prove that if x>2 then $x^4 - 8x^3+24x^2-32x+16$.

Your question looks incomplete. Is that supposed to be $x^4 - 8x^3+24x^2-32x+16 > 0$?

What's $(x-2)^4$? How does that help in your question?

 

[QuarkCharmer]

Thank you for the reply,

Sorry, yes the whole equation should be greater than zero, or equal to/less than something, I just made the question up as an example.

I see that (x-2)^4 is in fact equal to that polynomial. Therefor, for it is always going to be greater than zero because it is to an even power (and there is the case where x is 2, and then it IS equal to zero). I don't know how I am supposed to write this though? I'm aware that the conclusion is true for all x, but how would I even prove that:

$(x-2)^4 > 0$, for some x$\in$R

Edit:

Oh wait, I think I see what you are getting at...

 

[Curious3141]

Thank you for the reply,

Sorry, yes the whole equation should be greater than zero, or equal to/less than something, I just made the question up as an example.

I see that (x-2)^4 is in fact equal to that polynomial. Therefor, for it is always going to be greater than zero because it is to an even power (and there is the case where x is 2, and then it IS equal to zero). I don't know how I am supposed to write this though? I'm aware that the conclusion is true for all x, but how would I even prove that:

$(x-2)^4 > 0$, for some x$\in$R

That should be $(x-2)^4 \geq 0$ for some $x \in \mathbb{R}$. Equality occurs at x = 2. These little details are important.

You don't even need to bother with the case $x \leq 2$ since the question didn't ask for it.

Sketch the curve, and just make the appropriate arguments.

To prove the function is always increasing beyond x = 2, what can you say about the derivative for any x greater than 2?