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Factoring polynomials with five terms...

Posted on 2007-04-05
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Last Modified: 2012-06-27
Ok, going to try this again... I'd posted a similar Q once but figured it out (maybe I'll be lucky and figure this one out after posting)... anyway...
First off, this is NOT homework, or more specifically it's homework I've assigned myself so I can learn this.

I know how to remove greatest common denominators, and I know how to factor trinomials, and I know how to group to allow factoring...

In this problem:
f(x) = 2x^4 + 14x^3 + 25x^2 - 4x - 28

I know that I can group the first two and the last two and go from here:
2x^3(x + 7) + 25x^2 - 4(x + 7)
to here:
(2x^3 - 4)(x + 7) + 25x^2

But since the stated intent of this problem is "State all possible rational zeros" and I assume that means values for x such that the end result IS zero, I'm stumped as to how to get from my last point to THAT point...
All the things I've found about "grouping" are based on grouping 2 things together.  

Understand, I'm interested in knowing HOW to do this, not the answer to this one specifically...

Thanks.
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Question by:dagesi
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20 Comments
 
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Accepted Solution

by:
ozo earned 250 total points
ID: 18860611
factoring a general fourth order polynomial can get pretty complicated
http://mathworld.wolfram.com/QuarticEquation.html
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by:InteractiveMind
InteractiveMind earned 125 total points
ID: 18860655
f(x) = 2x^4 + 14x^3 + 25x^2 - 4x - 28

Go through a few integers close to 0, to see if any give f(that number)=0.

I tried from -3, then -2, ...

And found that f(-2)=0. So factor out (x+2):

f(x) = (x+2)(2x³+10x²+5x-14)

Use a similar method on g(x)=2x³+10x²+5x-14, you should see again that g(-2)=0 as well. So factor that from g(x):

f(x) = (x+2)(x+2)(2x²+6x-7) = (x+2)²(2x²+6x-7)

...Can you solve it from there?
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LVL 53

Assisted Solution

by:Infinity08
Infinity08 earned 125 total points
ID: 18860737
For this example, you can easily find one zero :

      (2x^3 - 4)(x + 7) + 25x^2

namely x = -2

So, you get :

      (x + 2)(2x^3 + 10x^2 + 5x - 14)

Once more, you can easily spot that x = -2 is again a zero :

      (x + 2)(x + 2)(2x^2 + 6x - 7)

Then we have a quadratic which can easily be solved :

      (-6 +/- sqrt(36 + 56)) / 4  =  (-3 +/- sqrt(23)) / 2

So we get :

      2(x + 2)(x + 2)(x + (3 + sqrt(23))/2)(x + (3 - sqrt(23))/2)



There's no real rule that always works - sometimes obvious zeros can be seen, which can help you further like in this case.

What can help you is to recognise general rules like :

      a^2 - b^2 = (a - b)(a + b)
      (a + b)^2 = a^2 + 2ab + b^2
      (a - b)^2 = a^2 - 2ab + b^2
      a^3 + b^3 = (a + b)(a^2 - ab + b^2)
      a^3 - b^3 = (a - b)(a^2 + ab + b^2)
      etc.

Note that a and b can be anything : x, 5, x^2, etc.

It's mostly just a question of recognising patterns ...
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by:dagesi
ID: 18860751
>ozo...
I've only glanced at that link so far but is it supposed to be easy to understand because it doesn't seem to be... (remember, I only glanced though)...

>InteractiveMind...
Is there a specific reason you picked the numbers to test you did or were you just trial and erroring...?

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LVL 84

Expert Comment

by:ozo
ID: 18860786
As I said, factoring fouth order polynomials in generalis not easy.
In some special cases, you can get lucky and find simple factors.
Newton-Raphson can find roots numerically
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LVL 84

Expert Comment

by:ozo
ID: 18860842
If you are oly looking for rational roots, for polynomials with integer coefficients, then there are a limited number of possibilities to try:
http://www.mathwords.com/r/rational_root_theorem.htm
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Expert Comment

by:InteractiveMind
ID: 18860858
> Is there a specific reason you picked the numbers to test you did or were you just trial and erroring

I used trial and error. The reason for this, is that usually if you are asked to solve a polynomial f(x)=0 in an exam, for example, then they usually make the factors fairly simple, such that a quick trial and error will allow you to solve it (as I did above).

Obviously, in the real world, polynomials rarely work out so easily - in which case, ozo's general solution would be the root you'd take..
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by:InteractiveMind
ID: 18860871
Wow, I had never realised that "rational root theorem" before!
Thanks for that ozo :-)
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by:Infinity08
ID: 18860902
>> Is there a specific reason you picked the numbers to test you did or were you just trial and erroring...?

It's partly guessing, partly deducing :

      f(x) = 2x^4 + 14x^3 + 25x^2 - 4x - 28

Since this is an exercise, there is most likely an easy solution, so since we can't spot a grouping at first sight, we'll start trying integer values.

Notice that positive values don't have to be checked, since there aren't enough negative terms to "counter" the positive terms. The 0 can be easily discarded too, since we have a constant term.
So, we try -1 which doesn't work. Then we try -2 ... and that one works.


It's not merely guessing, but it comes pretty close :)

As ozo said : luck often plays an important role too.
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Expert Comment

by:ozo
ID: 18860991
If you're looking for integer roots of
2x^4 + 14x^3 + 25x^2 - 4x - 28
the rational root theorem tells you that you only need to try factors of 28
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by:dagesi
ID: 18861730
Ok, I'll take your guys word for it that what you've said is the right way (so as not to make you sit around waiting for the points while I get the chance to try to understand all of that... =]
Thanks for the help...

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by:Infinity08
ID: 18862498
"guessing" is a valid technique in mathematics if it gives you the solution faster than the other techniques. It might sound a bit weird or "un-professional" at first, but in this case, it's the easiest way :)
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by:dagesi
ID: 18865245
I know for simple trinomials, I can almost know it just by looking... like:
 4x^2 + 12xy + 9y^2
Not meaning it has to be that simple but the idea in general...
I just wasn't sure whether there was something I was missing... As I said, I was reading about grouping and thought, maybe in this case, the idea was to group one group of two and one group of three...
I'm guessing that could be the way to go sometimes but clearly this wasn't one of those times.  =]
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by:dagesi
ID: 18865324
>ozo...
I don't suppose you could explain how that link you sent would work in this case, could you...?
I've been looking at it and it's just not making any sense whatsoever...
I mean it starts with
z^4 + a[sub3]z^3 + a[sub2]z^2 + a[sub1]z + a[sub0] = 0

then the next thing is all x's and a's with subs
and then after the next formula they're using nothing but z's with subs and exponents...

Scary because it says algebra and I thought algebra was not that advanced...
=]

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by:Infinity08
ID: 18865408
>> I'm guessing that could be the way to go sometimes but clearly this wasn't one of those times.  =]

Indeed ... if you're lucky, you can easily spot a grouping. This time it wasn't that obvious ... but still doable if you can spot this grouping :

        (x^2 + 4 x + 4)(2x^2 + 6x - 7)

Which isn't entirely impossible :)
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by:dagesi
ID: 18865529
>Infinity08...
I'm not sure we're talking about the same thing or not...
I meant for instance having:
qa^4 + rb^3 + sc^2 + tb + u
and being able to break it into:
(qa^4 + rb^3 + tb) + (sc^2 + u)
kind of thing... and having that actually be obvious for a breakdown...
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by:Infinity08
ID: 18867897
>> and being able to break it into:
>> (qa^4 + rb^3 + tb) + (sc^2 + u)

That usually doesn't bring you a lot closer to factoring the polynomial. I say usually, because in certain cases it will help you :)

I know all of this sounds vague, but a lot of exercise can help you :)
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by:dagesi
ID: 18869754
>Infinity08...
Yeah, from what you lot are saying it's a lot like just doing normal trinomials... enough practice and some of them become almost obvious at first look...
Though I love math, it SO should not involve guessing... =]
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by:Infinity08
ID: 18869782
>> Though I love math, it SO should not involve guessing... =]

The problem is that the alternative to "guessing" in this case involves almost brute-force like operations (which isn't really better than guessing).

If you are lucky, you can see a pattern, and use that to solve the problem without guessing ... but usually, some level of guessing is needed.

Oh well, you get it :)
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Expert Comment

by:ozo
ID: 18870239
There can be general principles like the rational root theorem that can help you guess intelligently.
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