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Springs
Registered Member
Date Joined Jul 2001
Total Posts : 636
| Posted 5/3/2003 4:56 PM (GMT -6) |   | x plus one, to the third power, times x,
minus two, times x minus one, to the second power, times x squared
plus x cubed, times x plus one.
Is how you could verbalize this expression:
[(x+1)^3]x - 2[(x-1)^2]x^2 + x^3(x + 1)
When you factor out the term (x + 1)x
you get (x + 1)x [ {x + 1)^2(1) - 2(x-1)x + x^2(1) ]
If you then simplify
(x+1)x [ x^2 + 2x + 1 -2x^2 - 2x + x^2 ]
then combine like terms:
(x+1)x [ x^2 + x^2 + (-2x^2) equals 0
and +2x + (-2x) equals 0
so there is only the +1 left
which means the answer is (x+1)x times +1 or (x+1)x
If you got that, can you factor this?
y^3 - 3y^2 - 4y + 12
Am I further out than the regular philosophy, poetry and politics?
aight.
Speaking of numbers, is Bennett a hypocrite for writing and selling "The Book of Vitues" while losing millions of dollars at gambling?
Springs
"Dreams are illustrations...from the book your soul is writing about you." -Marsha Norman
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Tom
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Date Joined Jul 2001
Total Posts : 10353
| Posted 5/3/2003 5:11 PM (GMT -6) | | I'll have an answer for you in a few hours. I'll have to wait for my daughter to wake up!
Tom
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Springs
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Date Joined Jul 2001
Total Posts : 636
| Posted 5/3/2003 5:23 PM (GMT -6) | | Cool!
One more thing, just in case... I want to know HOW to factor it.
I already know the answer, which is (y+2)(y-1)(y-3)
I just don't know what the process in between looks like.
Sheesh you get up early!
-S
"Dreams are illustrations...from the book your soul is writing about you." -Marsha Norman
Edited by - Springs on 05/03/2003 10:23:38 PM | | Back to Top | | |
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Tom
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Date Joined Jul 2001
Total Posts : 10353
| Posted 5/3/2003 5:38 PM (GMT -6) | | The dog woke me up barking. Our NEW dog. Our SECOND dog. The one I'm going to kill...
I've tried to get onto Messenger to say hi, but it's not logging me on.
I'll get you the factoring steps asap.
Tom
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Springs
Registered Member
Date Joined Jul 2001
Total Posts : 636
| Posted 5/3/2003 6:06 PM (GMT -6) | | And you wonder why you don't mourn their passing? {:P]
It would have been great to catch up on the IM. I have been studying way too much. I have no life.
I'm going to get some shut eye, and no doubt have exponential dreams.
Springs
"Dreams are illustrations...from the book your soul is writing about you." -Marsha Norman
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¡Que Payaso!
Registered Member
Date Joined Jul 2001
Total Posts : 108
| Posted 5/3/2003 9:08 PM (GMT -6) | | Springs,
As to the HOW part, I've only factored a couple of these in my lifetime, but the basic rule for starting out seems to be: GUESS. You try out something easy like (y-1) or (y+1) and divide using polynomial long division. If it's correct, you'll get a nice polynomial of degree 2 that you can UN-FOIL.
As for a formula (like the quadratic), you can check in book by William Dunham entitled Journey Through Genius ( I THINK he spells out the derivation ?) but it gets pretty messy.
If you can wait, I'll look it up after this weekend and post the formulas for you next week if you still need them.
Payaso 
"I don't know half of you half as well as I should like; and I like less than half of you half as well as you deserve." - Bilbo Baggins | | Back to Top | | |
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¡Que Payaso!
Registered Member
Date Joined Jul 2001
Total Posts : 108
| Posted 5/3/2003 9:26 PM (GMT -6) | | Springs,
Found this at http://mathforum.org/library/drmath/view/58544.html
About 1/3 of the way down the page, he arrives at a polynom deg 3 which looks like this:
x^3 + 7x^2 + 16x + 12
He goes on to factor it this way (the rest of this note is copied directly from the site):
*******************************************************************
NOW comes the tricky part: the factoring.
There are two methods:
trial and error (which means, you try a couple of likely
multiplications)
and
division.
I tried trial and error for a while but I didn't come up with an
answer.
So then I tried division.
Maybe you haven't seen division before in this context: it's pretty
similar to division with plain old numbers, but much cooler.
It looks something like this.
I think that our polynomial (x^3 + 7x^2 + 16x + 12) is going to be
divisible by x+2 (since we multiplied out some powers of x+2 and
added them)
So I try this
___________________________
x+2 |x^3 + 7x^2 + 16x +12
first,
I look at x^3 + 7x^2. How many times will x+2 go into x^3 + 7x^2?
I guess that it's around x^2 times, since x+2 times x^2 = x^3 + 2x^2
So now I write that down and subtract:
x^2
_________________________
x+2 |x^3 + 7x^2 + 16x +12
-(x^3 + 2x^2)
___________
5x^2
Next I bring down the next term:
x^2
___________________________
x+2 |x^3 + 7x^2 + 16x +12
-(x^3 + 2x^2)
_____________
5x^2 + 16x
and guess again:
This time, my guess is 5x. Can you see why?
x^2 + 5x
___________________________
x + 2 |x^3 + 7x^2 + 16x +12
-(x^3 + 2x^2)
__________
5x^2 + 16x
-(5x^2 + 10x)
_________
6x
Next, I'll bring down the 12 and finish up with a guess of 6.
x^2 + 5x + 6
_______________________________
x+2|x^3 + 7x^2 + 16x +12
-(x^3 + 2x^2)
_____________
5x^2 + 16x
- (5x^2 + 10x)
______________
6x +12
-(6x +12)
_________
0
Whew! Looks like it worked... We can check by multiplying
(x+2) (x^2 + 5x +6)
Now we've factored x^3 + 7X^2 + 16x +12
To FULLY factor, we'll need to see if either x+2 or x^2 + 5x +6
have factors.
********************************************************************
Payaso
"I don't know half of you half as well as I should like; and I like less than half of you half as well as you deserve." - Bilbo Baggins | | Back to Top | | |
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Springs
Registered Member
Date Joined Jul 2001
Total Posts : 636
| Posted 5/4/2003 8:37 AM (GMT -6) | | Hey Tom,
My 17 year old is a math wiz too. We really should get them together. He helped me with the quadractic (second degree) polynomials. We have been happily factoring them to result in the product of two bi-nomials. But the cubic or third degree polynomials with four terms are closer to calculus and a bit out of his league.
Payaso! Thank You! That is a great website. I really appreciate your help. I'm not sure I'll put that book on my summer reading list, but it sounds interesting. I love learning this stuff till I get stuck. 
"Dreams are illustrations...from the book your soul is writing about you." -Marsha Norman
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Tom
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Date Joined Jul 2001
Total Posts : 10353
| Posted 5/4/2003 9:41 AM (GMT -6) | | Sorry I didn't get you an answer. My girl has been out all day (choir, friends, and now staying over night) so I haven't even seen her! Math is definitely not my field!
Tom
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Springs
Registered Member
Date Joined Jul 2001
Total Posts : 636
| Posted 5/4/2003 10:39 AM (GMT -6) | | My teenager is on the go too. Every time he runs by I ask him a math question. He may know more than he's letting on.
"Dreams are illustrations...from the book your soul is writing about you." -Marsha Norman
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Magneton
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Date Joined Aug 2001
Total Posts : 132
| Posted 5/4/2003 12:55 PM (GMT -6) | | Springs,
I agree with Payaso that guessing is involved, but you have some clues to guide your guesses.
y^3 - 3y^2 - 4y + 12
The +1 coefficient on y^3 tells you that the y coefficient on each term will be +1.
The +12 term tells you that the integer terms on the 3 factors will multiply to 12, i.e. they can be (2, 2, and 3) or (4, 3, and 1) or 6, 2, and 1). Their signs could be (all +) or (one + and two -). The negative coefficients on y^2 and y tell you there must be some negative coefficients in the factors, which rules out the (all +).
(y-2)(y+2)(y-3) meets these requirements, and is the correct answer. Whether you find the factors all together like this, or one at a time is up to you. This one is easy to verify since multiplying the 1st two factors gives (y^2 – 4)(y-3).
Magneton
“If God speaks to man, he undoubtedly uses the language of mathematics.” Henri Poincare
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GregBoyd
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Date Joined Jul 2001
Total Posts : 1818
| Posted 5/4/2003 1:01 PM (GMT -6) | | How come God never gives us a "word of knowledge" when we need it, heh?
Lets talk the problem of evil -- its easier than all this alegra stuff.
Doc
dang we got some smart people on this board!
"The responsibility is with the chooser -- God is blameless." Plato, Republic, 10. | | Back to Top | | |
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Magneton
Registered Member
Date Joined Aug 2001
Total Posts : 132
| Posted 5/4/2003 3:44 PM (GMT -6) | | Doc: How come God never gives us a "word of knowledge" when we need it, heh?
Mag: I suppose we need to do the homework so we understand the language that the word of knowledge comes in. In the case of mathematics, Johannes Kepler said it well:
“The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.”
Springs, best wishes on our mathematical studies. Where are you headed with it?
Magneton
“Go deep, go far into the scheme of things, and there is mathematics everywhere. Go deeper, go nearer to the core of reality, and there is music everywhere.” Slichter, Science in a Tavern
Edited by - Magneton on 05/04/2003 8:45:16 PM | | Back to Top | | |
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Springs
Registered Member
Date Joined Jul 2001
Total Posts : 636
| Posted 5/4/2003 5:18 PM (GMT -6) | | Mags, thanks for the pointers. I knew that for second degree quadratics the factors in the last term added up to the second term, for help in guessing the right combination. x^2 = (r+s)x + rs But, I did not know this neat trick for the third degree polynomials. Nice.
I have to pass Algebra in order to complete my bachelors, by the end of this month. oy.
Doc, necessary contingencies and might counterfactuals are easier than this?
Springs
"Dreams are illustrations...from the book your soul is writing about you." -Marsha Norman
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Starkman
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Date Joined Oct 2001
Total Posts : 1747
| Posted 5/6/2003 1:21 AM (GMT -6) | | x plus one, to the third power, times x,
minus two, times x minus one, to the second power, times x squared
plus x cubed, times x plus one.
Is how you could verbalize this expression:
[(x+1)^3]x - 2[(x-1)^2]x^2 + x^3(x + 1)
When you factor out the term (x + 1)x
you get (x + 1)x [ {x + 1)^2(1) - 2(x-1)x + x^2(1) ]
If you then simplify
(x+1)x [ x^2 + 2x + 1 -2x^2 - 2x + x^2 ]
then combine like terms:
(x+1)x [ x^2 + x^2 + (-2x^2) equals 0
and +2x + (-2x) equals 0
so there is only the +1 left
which means the answer is (x+1)x times +1 or (x+1)x
If you got that, can you factor this?
y^3 - 3y^2 - 4y + 12
I'm telling you, this site has become corrupted. Virus, I'd imagine! I say Format and reinstall! No, better yet, FDisk! (That'll be the last of them there darned characters corrupt this board!)
Starkman
Edited by - Starkman on 05/06/2003 06:22:32 AM | | Back to Top | | |
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Springs
Registered Member
Date Joined Jul 2001
Total Posts : 636
| Posted 5/6/2003 6:56 PM (GMT -6) | | Corrupt?! Virus like?! Preach it brother!
I quote: "College Alegebra" 2000 - "Although we are often unaware of its presence, mathematics permeates nearly every aspect of life in the modern world. With the advent of modern technology, mathematics has steadily been playing a greater role in our everyday life. In fact, you were probably awakended this morning by a digital alarm clock, made a phone call that used digital transmission, sent and e-mail message over the Internet, drove a car with digitally controlled fuel injection, listened to music on a CD player, then fell asleep in a room whose temperature is controlled by a digital thermostat. In each of these activities mathematics is crucially involved. In general, a property such as the intensity or frequency of sound, oxygen level in the exhaust emission from a car, the colors in an image,..."
yada, yada, yada
Computers in cars just mean the local mechanic can't help you. CD's last about a year! 8 Tracks lasted forever! Digital schmigital!
I got my ass kicked by algebra today! If I pass this course I hope to never lay eyes on an exponent again!
Sorry Mags. You know I love to learn, but cramming four hundred new rules into my head and then trying to apply them, in the a weeks time, is not learning, it's it's...It's making me crazy.
X squared to the...
Springs
"Dreams are illustrations...from the book your soul is writing about you." -Marsha Norman
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selah
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Date Joined Oct 2001
Total Posts : 1210
| Posted 5/7/2003 1:23 AM (GMT -6) | | Hey Springs,
Did I tell you I suffered through that class? When you get through we should burn our hundred dollar books together!
love,
selah
Then He showed me the river whose waters give life, sparkling like crystal, flowing out from the throne of God and of the Lamb. Rev. 22:1 | | Back to Top | | |
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Starkman
Registered Member
Date Joined Oct 2001
Total Posts : 1747
| Posted 5/7/2003 1:51 AM (GMT -6) | | See, you're smart, Springs. You're already aware of the virus's ability, and you've got your anti-alge . . . anti-virus software working!
Good for you. X can just go exponent Y till the moon turns red, as far as I'm concerned. (Have to admit, though, algebra looks like it'd be really fun and interesting IF there was any reason on God's green earth that I would ever use it; not someone else using it, ME! And that ain't gonna happen in this life time!)
Starkman
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Springs
Registered Member
Date Joined Jul 2001
Total Posts : 636
| Posted 5/7/2003 4:47 AM (GMT -6) | | Thanks Selah,
Future pacing toward an end goal has always been a great motivational strategy for me. Course I'm a bibliophile so I'm not sure I could burn a book. Hey, I know, how about me and you and Hose get together with some salt and some key limes. 
Starkman,
You are right, it is very intriguing. I really do enjoy it when I am able to figure it out. It's like an intricate puzzle. Understanding Algebra would undoubtedly increase my ability to grasp complicated concepts in other fields. Which could be invaluable when my family dumps me off on a country road for my dereliction of duty around here. I don't have time to volunteer, visit my friends, or do housework (wait a minute...that's a good thing)I spend more time reading algebra cyberboards than I do this one, hmmm not sure which side of the equation the fits on...But today is a new day. I called my professor and told him there is no way I can get all the work done, I need to get done, in the next three weeks in order to graduate, and he responded, "You can get it done. You have time." So I am so encouraged now. Who knew? What a good deal.
If I decide to give up and become a panhandler at least I have learned enough to figure out how much change I'll need to make a hundred bucks. 5x + 10y + 25z = 100.
God bless Fibonacci and Mags,
Springs
"Dreams are illustrations...from the book your soul is writing about you." -Marsha Norman
Edited by - Springs on 05/07/2003 10:02:50 AM | | Back to Top | | |
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selah
Registered Member
Date Joined Oct 2001
Total Posts : 1210
| Posted 5/7/2003 6:35 AM (GMT -6) | | Springs,
Sounds like a plan--though I bet you'll be whistlin' a different tune after you experience the FINAL!!  (how's that for a future pace?) How about I save my matches and most holy offering spot until I hear back.
love
selah
Then He showed me the river whose waters give life, sparkling like crystal, flowing out from the throne of God and of the Lamb. Rev. 22:1 | | Back to Top | | |
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Laplace
Registered Member
Date Joined Apr 2003
Total Posts : 138
| Posted 5/15/2003 10:00 AM (GMT -6) | | Agree this entire thread is QUITE off the topic, however, I'd like to mention a few things:
- Algebra is the "alphabet" of all higher mathematics (Calculus, Statistics, etc.) and subsequently, all higher sciences and engineering
- There is a theorem that exists that all REAL (non-imaginary) solutions to any polynomial can be found by finding a subset of the quotient of factors of the highest degree divided by the lowest degree or vice-versa
- The Fundamental Theorem of Algebra was first proved by Karl Friedrich Gauss (Germany) for his PhD Dissertation. It proved that any polynomial to the n-th degree has n-roots (real or imaginary, included repeated roots).
- Algebra in terms of pure mathematical analysis however is a much harder beast.
"Balance in all things" - Dak'kon, Planescape Torment | | Back to Top | | |
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