# Partition Numbers: Approximating & Solving Inequalities

• undrcvrbro
In summary, Dick was able to help me find the roots of the function f(x) and he advised me to test points inside the intervals to see what the sign is.
undrcvrbro

## Homework Statement

Use a graphing utility to approximate the partition numbers of the function f(x) to two decimal places. Then solve the following inequalities.
(a) f(x)>0
(b) f(x)<0
Express all answers in interval notation

## The Attempt at a Solution

The partition can be calculated by just finding the max and the min of the graph with my TI-83 Plus, right? It's solving the inequalities I'm having trouble with. What exactly is it asking for?

What are the 'partition numbers' of a function f(x)?? Can you define that??

wow..sorry

brain fart. I'm a little flooded in work and lacking sleep..sorry.

the function is f(x)=x^3-3x^2-2x+5

undrcvrbro said:
brain fart. I'm a little flooded in work and lacking sleep..sorry.

the function is f(x)=x^3-3x^2-2x+5

I didn't mean define the function, I meant define the term 'partition numbers' - it's not a term I've seen before. If you want to find where f(x)>0 and f(x)<0 then you generally want to find the roots first, values of x such that f(x)=0. Are those 'partition numbers'??

Dick said:
I didn't mean define the function, I meant define the term 'partition numbers' - it's not a term I've seen before. If you want to find where f(x)>0 and f(x)<0 then you generally want to find the roots first, values of x such that f(x)=0. Are those 'partition numbers'??

Yes, partition numbers are values of x such that f(x)=0. So then once I have found those, what should I do in order to find f(x)>0 and f(x)<0?

Once you've found those, you've found the only places where f(x) can change sign. So if the roots are a<b<c, then f(x) has a constant sign on the intervals x<a, a<x<b, b<x<c and x>c. To figure out what that sign is, just test a point inside each of the intervals.

ah, okay. Thanks a lot Dick, I appreciate your help!

## 1. What are partition numbers?

Partition numbers refer to the number of ways a positive integer can be expressed as a sum of positive integers. For example, the partition numbers of 4 are 5, as 4 can be expressed as 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.

## 2. How can we approximate partition numbers?

There are various methods for approximating partition numbers, such as using generating functions, recursion, or even using computer algorithms. However, the most common and efficient method is using Euler's Pentagonal Number Theorem.

## 3. What is Euler's Pentagonal Number Theorem?

Euler's Pentagonal Number Theorem states that the number of partitions of an integer n is equal to the sum of the partitions of n-k, where k is a generalized pentagonal number. This theorem provides a recursive formula for calculating partition numbers, which can be used for approximation.

## 4. How can partition numbers be used to solve inequalities?

Partition numbers can be used to solve inequalities by approximating the number of solutions to the inequality. For example, if we have an inequality such as x+y+z<10, we can use partition numbers to approximate the number of solutions for different values of x, y, and z, giving us an idea of the range of solutions for the inequality.

## 5. Are there any practical applications of partition numbers?

Yes, partition numbers have various practical applications in fields such as number theory, combinatorics, and computer science. They can be used to solve problems involving distributions and combinations, and have also been used in cryptography and coding theory.

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