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In Part II of this article, we are going to examine a sure-fire way to factor non-monic trinomials. We are going to examine the proof which gives us this powerful method. After you are done with this presentation, you will not only be able to factor any non-monic, you will understand the reason why this method works.

To begin our presentation, let us take an arbitrary non-monic trinomial: to do this we need to introduce the constants a, b, and c. Thus our generic non-monic becomes ax^2 + bx + c, in which a is an integer greater than 1. Our end aim is to reverse FOIL this so that we can factor it into the form (dx + e)*(fx + g), where d, e, f, and g are integers. If you now FOIL this product, you obtain dfx^2 + dgx+ efx + eg, which we can write as dfx^2 + (dg + ef)x + eg, since the common factor of both dg and ef is x.

If we compare this last expression with the original non-monic, we that df must be equal to a; dg + ef must equal b and eg must equal c. If this were not the case, we would have a factored expression which is incorrect. Now this is where some thought is required, so I enjoin you to put your thinking cap on and follow closely. Feel free to reread as necessary to make sure you are following the argument.

Our goal is to determine d,e,f, and g. To do this we use a clever trick. We multiply a and c together. We note that this must equate to dfeg, since a = df and c = eg. Now remember that the middle coefficient, or b, is equal to dg + ef. If we break ac down into all its component factors, and observe which sum is equal to b, then we have found dg and ef. If we then divide a, which is df by dg, we get df/dg = f/g, since the d's cancel; similarly if we divide ef/df = e/d. We now have all four unknown letters solved, namely d,e,f, and g.

Because the last part was cryptic and probably somewhat hard to follow at first blush, I am going to do a specific example which will make this whole argument clear. Let us take the non-monic trinomial 6x^2 + 17x + 5. Let us multiply a*c or 6*5 = 30. Now break down 30 into all of its possible factor pairs: 1 30, 2 15, 3 10, and 5 6. Find the factor pair which when added or subtracted yields the middle coefficient 17. Clearly this is the pair 2 15. Since b = dg + ef, we know that 2 is dg and 15 is ef. If we divide 2 by 6 (which is dg by df) we get 1/3, which tells us that g is 1 and f is 3; similarly if we divide 6 by 15 and reduce to lowest terms, we get 6/15 or 2/5, which tells us that 2 is d and 5 is e. Thus the factorization is (dx + e)*(fx + g) or (2x + 5)*(3x + 1). If you FOIL this, you will see indeed that you get back the original non-monic.

To make this procedure clear, let us do another. Let us factor the non-monic 6x^2 + 41x + 70. We multiply 6*70 = 420. The factor pairs of 420 are 1 420, 2 210, 3 140, 4 105, 5 84, 6 70, 10 42, 14 30, and 20 21. The pair which added or subtracted produces 41 is 20 21. Thus 20 is dg and 21 is ef. When we divide both of these by a which is df and reduce, we have 20/6 = 10/3 = g/f; and 21/6 = 7/2, which is e/d. Thus 6x^2 + 41x + 70 can be factored into the form (dx + e)*(fx + g) = ( 2x + 7)*(3x + 10). Now how's that for sure-fire!

Note that we have not considered here cases of non-monics which involve negative numbers. Such is the case with 6x^2 + x -70. The method, albeit with a slight adjustment, works just as well. Once you have the method down for the cases in which all the numbers are positive, applying it to cases in which either b or c or both are negative, is academic.

Joe is a prolific writer of self-help and educational material and is the creator and author of over a dozen books and ebooks which have been read throughout the world. He is a former teacher of high school and college mathematics and has recently returned as a professor of mathematics at a local community college in New Jersey.

Joe propagates his Wiz Kid Teaching Philosophy through his writings and lectures and loves to turn "math-haters" into "math-lovers." See his website http://www.mathbyjoe.com for more information and for testimonials, and try out one of his ebooks here http://www.mathbyjoe.com/page/page/2924777.htm to achieve better grades in math.

Bling Cases for the Holidays: The Perfect Gift

Looking for that perfect holiday gift? A gift that can be used on a daily basis and is both inexpensive and hip at the same time? Well, you have met your match. Rhinestone and crystal cell phone cases have become extremely popular throughout the world, and this holiday season many will flock to cell phone stores and websites to buy phone cases. Customized cases offer cell phone users a uniqueness that cannot be found elsewhere.

Some may ask: what is a bling case? To help clarify everyone's questions, bling cases are rhinestone cell phone cases or crystal cases, that either have stones or shiny surfaces on their exterior. The cases are crafted with a wide variety of designs to showcase a plethora of interests including dog paws, leopard prints, butterflies, and everything else one can think of. With a unique and cool case, your phone will be transformed from being one in 10 million+ to a prized possession.

Think about it just for one second: mobile phones are everywhere. I have one, you have one, our great-grandmothers have them, but what's to distinguish yours from your girlfriends? Surely she would appreciate one more thing to make her feel special.

The holiday season presents more than a few worries. If you live in an area that snows, you will surely need to think about getting up early enough to scrape snow off of the car. If you have a holiday party, you will need to figure out who is going to bring the jello mold, and, perhaps even more nerve-wracking, you will need to find gifts for everyone. This holiday season, when looking for presents, I highly suggest you cross that tast off your list and consider a rhinestone cell phone case for your friends or family members, especially for the ladies. Why? See below:

1) Cost-effective

2) Useful on a daily basis

3) Unique

Good luck with your gift hunt, and may the force be with you.

Sincerely,

BlingCases.com

About the Author

Bling Cases is the leading manufacturer and retailer of high-quality Rhinestone Phone Cases and Crystal Phone Cases. With a wide variety of styles, colors, and stones, BlingCases.com is sure to have what you're looking for.

I have decided to convert to Christianity and part time to convert to ISLAM, just in case?

Christianity says you must believe in Jesus as your savior or BURN in hell, Islam says you must believe in Allah and Mohamed or burn in hell.
Since these two and ZOROASTRIANISM(where they took hell from) are the religions that believe in hell, I tought I would convert to both just in case, so NO BURNING AND TORTURING FOR ME.
Was it a good choice, or should I have chosen rationality and throw in the GARBAGE both beliefs?

don't forget the other things they took from zoroastrianism like the struggle between the "good" and the "evil" god aka "God" and "satan", satan having little more significance than the tempter who "caused" us to eat the fruit from the tree. Satan then turns into this powerful "fallen angel" struggling with god over domination of the world. whatever happened to god being in complete control. what happened to ALL things coming from god, both the good and the bad. what happened to the great supreme being. how dare the christians insult the great power of god with this puny little "devil". and how DARE they even CONSIDER that this little devil has absolutely ANY INFLUENCE OVER ANYTHING IN MY WORLD! its enough to make you want to wipe out a civilisation or two off the face of the earth to remind them your in control... only they would blame it on satan... CURSE THEM!!
=)

Doe trial continues as the judge kills district's motion to dismiss the case
The judge in the LMSD vs. Doe case rejected yet another motion by the school district's attorney to have the suit against it dismissed Monday. The district's latest attempt to have the case thrown out came Monday afternoon directly after the plaintiffs rested their case.

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