I have ansers all you need to know how to get answers in step by steop explanation all math work explain me too. ud+1. Prove the following polynomial is e(n). Use algebraic manipulations.P(n)=5n+20 n³ - 10 n² + 10(a) Prove 0(n*).(b) Prove (n*).Therefore,43P(n) = 5n²+ 20n - 10 m² +102<5n4+201³+10243P(n) = 5n²+ 20 m²3<5n4 +201³ (^_) + 10 (212) 4101)< n²4 (5+02 + 0.000000,n4 (5-0.1)4> 4.9 nof 64.9 nM2lcv1OD2> 5n²4-10 n²22 nΣ 5n²-101² (70)² 7210, (+)2)>1010 m² +10m310< P(n) < 5.2001 n'Discard Neg Termsm2100, (n))Note thistransformatiomay be appliedonly to pasDiscord Pos TerusThismanbe applied)to only Nag terms,m> 100

Answered: 1. Prove the following polynomial is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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I have ansers all you need to know how to get answers in step by steop explanation all math work explain me too.

ud
+
1. Prove the following polynomial is e(n). Use algebraic manipulations.
P(n)=5n+20 n³ - 10 n² + 10
(a) Prove 0(n*).
(b) Prove (n*).
Therefore,
4
3
P(n) = 5n²+ 20n - 10 m² +10
2
<5n4
+201³
+10
2
4
3
P(n) = 5n²+ 20 m²
3
<5n4 +201³ (^_) + 10 (212) 4
101)
< n²4 (5+02 + 0.000000,
<m² (52001)
>n4 (5-0.1)
4
> 4.9 n
of 6
4.9 n
M2lcv
1
OD
2
> 5n²4-10 n²
2
2 n
Σ 5n²-101² (70)² 7210, (+)2)
>
10
10 m² +10
m310
< P(n) < 5.2001 n'
Discard Neg Terms
m2100, (n))
Note this
transformatio
may be applied
only to pas
Discord Pos Terus
This
man
be applied)
to only Nag terms,
m> 100

Expert Solution

Step 1: Step 1

To show that: $P (n) = O (n^{4})$

We will use the definition of big oh notation to show this, we will find a positive constant (say C) and a real number N such that $P (n)$ is bounded above by $C n^{4}$ whenever $n \geq N$ .

To show: $P (n) \leq C n^{4} whenever n \geq N$ .

$\begin{aligned} P (n) & = 5 n^{4} + 20 n^{3} - 10 n^{2} + 10 \\ \leq 5 n^{4} + 20 n^{3} + 10 \end{aligned}$

In the above step, $- 10 n^{2}$ term will be always negative, so we have removed that to get an upper bound. Next we will show that $20 n^{3}$ and 10 terms is dominated by some constant times n⁴ .