My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Thanks Tree6Thanks
Reply
 
LinkBack Thread Tools Display Modes
October 16th, 2018, 06:40 PM   #1
Senior Member
 
SenatorArmstrong's Avatar
 
Joined: Nov 2015
From: United States of America

Posts: 181
Thanks: 21

Math Focus: Calculus and Physics
Stuck on finding solutions

How do I factor z^5 - 3z^4 - 16z + 48 = 0?

I was thinking rational root theorem. But when I look at the solution it uses three back to back iterations of synthetic division, and I got slightly confused. Any tips?
SenatorArmstrong is offline  
 
October 16th, 2018, 06:47 PM   #2
Senior Member
 
ProofOfALifetime's Avatar
 
Joined: Oct 2016
From: Arizona

Posts: 139
Thanks: 28

Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
Hi SenatorArmstrong, you should focus on what you can factor out of the first two terms. Then, can you factor something out of the last two terms? Try it out! (Hint: you'll need to factor as much as possible, so after your first round of factoring, check to see if any of the factors can be factored further)
Thanks from SenatorArmstrong

Last edited by ProofOfALifetime; October 16th, 2018 at 06:50 PM.
ProofOfALifetime is offline  
October 16th, 2018, 06:48 PM   #3
Senior Member
 
ProofOfALifetime's Avatar
 
Joined: Oct 2016
From: Arizona

Posts: 139
Thanks: 28

Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
Let me know how it works out, I can help more if you need.
ProofOfALifetime is offline  
October 16th, 2018, 06:52 PM   #4
Senior Member
 
Joined: Oct 2009

Posts: 609
Thanks: 186

OK, you mentioned the rational root theorem. Very good start!
So if you apply the rational root theorem to this polynomial, what do they say the rational roots are?
Micrm@ss is offline  
October 16th, 2018, 06:56 PM   #5
Senior Member
 
ProofOfALifetime's Avatar
 
Joined: Oct 2016
From: Arizona

Posts: 139
Thanks: 28

Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
Quote:
Originally Posted by Micrm@ss View Post
OK, you mentioned the rational root theorem. Very good start!
So if you apply the rational root theorem to this polynomial, what do they say the rational roots are?
You all are making it more complicated than need be. A simple factor by grouping works, and then factoring again.
ProofOfALifetime is offline  
October 16th, 2018, 08:06 PM   #6
Senior Member
 
ProofOfALifetime's Avatar
 
Joined: Oct 2016
From: Arizona

Posts: 139
Thanks: 28

Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
Quote:
Originally Posted by ProofOfALifetime View Post
You all are making it more complicated than need be. A simple factor by grouping works, and then factoring again.
I was just kidding
ProofOfALifetime is offline  
October 16th, 2018, 08:09 PM   #7
Senior Member
 
Joined: Oct 2009

Posts: 609
Thanks: 186

Quote:
Originally Posted by ProofOfALifetime View Post
You all are making it more complicated than need be. A simple factor by grouping works, and then factoring again.
You are definitely right. I missed that. It is indeed a lot simpler. But these are tricks that don't always work and where you need to be quite lucky. A rational root theorem is much more generally applicable and gives you all the rational root. Of course, it still isn't general enough for the general quintic.
Micrm@ss is offline  
October 16th, 2018, 08:20 PM   #8
Senior Member
 
ProofOfALifetime's Avatar
 
Joined: Oct 2016
From: Arizona

Posts: 139
Thanks: 28

Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
Quote:
Originally Posted by Micrm@ss View Post
You are definitely right. I missed that. It is indeed a lot simpler. But these are tricks that don't always work and where you need to be quite lucky. A rational root theorem is much more generally applicable and gives you all the rational root. Of course, it still isn't general enough for the general quintic.
What tipped me off was that he said, "how do you factor" so I sort of made the assumption. I love the rational roots theorem. I especially like how it's used as an alternative way to prove $\sqrt{2}$ is irrational. It's fascinating. (I'm not much of a fan of the proof by contradiction method of $\sqrt{2}$ being irrational.)
ProofOfALifetime is offline  
October 16th, 2018, 08:22 PM   #9
Senior Member
 
SenatorArmstrong's Avatar
 
Joined: Nov 2015
From: United States of America

Posts: 181
Thanks: 21

Math Focus: Calculus and Physics
Quote:
Originally Posted by Micrm@ss View Post
OK, you mentioned the rational root theorem. Very good start!
So if you apply the rational root theorem to this polynomial, what do they say the rational roots are?
The rational room theorem tells me that $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8 ,\pm 12, \pm 16$ are potential solutions.

I began by plugging in these values starting with $\pm 1$ when doing so, those values made the equation false.

The value of 2 worked. I then went ahead with synthetic division.

I went from $z^5 - 3z^4 - 16z + 48$ $\Rightarrow$ $z^4 - 5z^3 + 10z^2 - 20z + 24$

I used the rational root theorem again and found that $-3$ is a root.

Proceeding with synthetic division...

$z^4 - 5z^3 + 10z^2 - 20z + 24$ $\Rightarrow$ $z^3 - 2z^2 + 4z - 8$

I factored this by method of grouping.

$z^2(z-2) + 4(z-2)$ $\Rightarrow$ $(z^2 + 4)(z-2)$

I've been able to conclude that $z=2, -3, \pm 2i$

My original polynomial is a fifth degree polynomial so I was expecting to find one more solution. Where did I go wrong?

Thanks a lot for the help.

Kind Regards

Last edited by SenatorArmstrong; October 16th, 2018 at 08:25 PM.
SenatorArmstrong is offline  
October 16th, 2018, 08:25 PM   #10
Senior Member
 
SenatorArmstrong's Avatar
 
Joined: Nov 2015
From: United States of America

Posts: 181
Thanks: 21

Math Focus: Calculus and Physics
Quote:
Originally Posted by ProofOfALifetime View Post
Hi SenatorArmstrong, you should focus on what you can factor out of the first two terms. Then, can you factor something out of the last two terms? Try it out! (Hint: you'll need to factor as much as possible, so after your first round of factoring, check to see if any of the factors can be factored further)
$z^5 - 3z^4 - 16z + 48 = 0$ $\Rightarrow$ $z^4(z-3) - 16(z+3)$

The $(z-3)$ and $(z+3)$ looks like a difference of two squares, but don't think I can apply that here. I'm curious how you can factor this quicker since rational root theorem can get time consuming.

I appreciate your help.
SenatorArmstrong is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
finding, solutions, stuck



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Finding the non-trivial solutions bobbo Algebra 8 October 30th, 2013 03:31 PM
finding solutions within the inverval [0,360) and live504 Algebra 6 April 8th, 2012 10:35 AM
Finding the Centroid of this triangle - stuck! grogmachine Algebra 3 August 5th, 2011 02:16 PM
Finding points and solutions. blackobisk Algebra 1 April 9th, 2009 03:28 PM
finding solutions within the inverval [0,360) and live504 Calculus 1 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.