An application of newtons sum

For a given polynomial of degree $n$ with roots $r_1, r_2, \ldots r_n$ , let $s_k$ be the kth symmetric sum and $p_k = \sum_{i=1}^n r_i ^k$ is the kth power sum

Claim: If $s_1=s_2=s_3=...=s_k=0$ ,then $p_1=p_2=p_3=.,..=p_k=0$ . where

Proof: By induction on $k$ . Base case: Since $s_1 = p_1$ , hence if $s_1 = 0$ then $p_1 = 0$ .

Induction step: Suppose that for some $j$ , we know that $p_1, p_2, \ldots p_j = 0$ . Then, by newtons sum, we have

$p_k=s_1p_{k-1}-s_2p_{k-2}+....+(-1)^{k-1}s_{k-2}p_2+(-1)^{k}s_{k-1}p_1+(-1)^{k+1}ks_k = 0 + 0 + \ldots + 0.$

Easy Math Editor

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There's no need to apply Newton's Sum.

If $s_1 = s_2 = \cdots = s_k = 0$ , then the roots satisfy the equation, $x^k = 0$ . So all its roots are 0, then by definition, $p_1 = p_2 = \cdots = p_k = 0$ .

Pi Han Goh - 4 years, 4 months ago

what about $x^{100}-x^2+x+1=0$ i meant for that

Aareyan Manzoor - 4 years, 4 months ago

I don't follow. What about that polynomial?

Pi Han Goh - 4 years, 4 months ago

@Pi Han Goh – look at this

Aareyan Manzoor - 4 years, 4 months ago

@Aareyan Manzoor – That one only applies for $s_1 = s_2 = \ldots = s_{98}= 0$ . And $s_{99} = -99 \ne 0$ .

Pi Han Goh - 4 years, 4 months ago

@Pi Han Goh – i meant that see, both variables are "n".

Aareyan Manzoor - 4 years, 4 months ago

@Aareyan Manzoor – I don't understand what you're saying.

Pi Han Goh - 4 years, 4 months ago

@Pi Han Goh – i gave wrong variables in note, just realized now. edited

Aareyan Manzoor - 4 years, 4 months ago

@Aareyan Manzoor Great note! I've cleaned it up so that it's much easier to understand what you are saying.

Presenting it via a proof by induction isn't necessary, but helps build the idea that all we need to show $p_j = 0$ is up to $s_j = 0$ .

By improving the format of your presentation, you make it immediately clear what you want to show, and why it's interesting.

Calvin Lin Staff - 4 years, 3 months ago

thank you sir!

Aareyan Manzoor - 4 years, 3 months ago

nice way....

Dev Sharma - 4 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$