   Chapter 11.2, Problem 25ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
1 views

# Suppose P ( n ) = a m n m + a m − 1 n m − 1 + ⋯ + a 2 n 2 + a 1 n + a 0 , where all the coefficients where all the coefficients a 0 ,   ​ a 1 ,   … ,   a m , are real numbers and a m > 0 . a. Prove that P ( n ) is Ω ( n m ) by using the general procedure described in Example 11.2.4. b. Prove that P ( n ) is O ( n m ) .c. Justify the conclusion that P ( n ) is Θ ( n m )

To determine

(a)

Prove that P(n) is Ω(nm)

Explanation

Given information:

P(n)=amnm+am1nm1+...+a2n2+a1n+a0

Concept used:

f is of order at least g, written f(n) is Ω(g(n)), if and only if there exists positive real numbers A and ar such that,

Ag(n)f(n) for every integer na

Proof:

Let m be a non-negative integer, let P(n) be a polynomial of degree m, and suppose the coefficient am of nm is positive.

To find big-omega for P(n): Let a0

A=12am let’s find a as follows,

The coefficient of the highest power is am

Sum of absolute values of coefficients is |am1|+|am2|+...+|a2|+|a1|+|a0|. Thus,

A=12am and a=2am×(|am1|+|am2|+...+|a2|+|a1|+|a0|)

Requiring na means that,

n2am×(|am1|+|am2|+...+|a2|+|a1|+|a0|)

And multiplying both sides by nm12 gives,

nm(| a m1|+| a m2|+...+| a 2|+| a 1|+| a 0|)×n m1am1am(| a m1|n m1+| a m2|n m1+...+| a 2|n m1+| a 1|n m1+| a 0|n m1)1am(| a m1|n m1+| a m2|n m2+

To determine

(b)

Prove that P(n) is O(nm)

To determine

(c)

Justify the conclusion that P(n) is θ(nm)

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