Possible flaw with the term "factor"

[spikerz66]

hello im new her this is only my second post so i dont know if i put this in the correct place or not but yeah...

 

my problem is that on a test today my teaches asked us to factor:

2(x)squared+13(x)+15

i factored it out and got (2x+10)(x+1.5) my reasoning is as follows

2(x squared+6.5x+7.5) i divided by 2

2(x+5)(x+1.5) factored the terms

(2x+10)(x+1.5) if you F.O.I.L. out the terms you get the original equasion (which i thought was "factoring" but however my teacher got a different answer)

 

after thinking over this for a while i came to the conclusion that the term "factor" should have been replaced with "find the simplest factor" because what i had was a factor. the only thing why i can think of my problem being wrong is that if you take any whole number, ill say 5 and divide it by a decimal, ill say 3.125432 you get another infinite number 1.59977884657............ so with that said you could in theory have an infiniter number of solutions to a single "factor" problem

 

here are some other problems that were on the test:

3x squared+21x

i got (x+0)(3x+21) which again checks out

 

2x squared-32x

i got (x+16)(x-16) again checking out

 

could someone please explain to me why my original problem isnt correct?

 

(by the way if anyone can tell me how to make "squared" that would also be helpfull)

 

-BLK

[timo]

(2x+10)(x+3/2) = 2(x+5)(x+3/2) = (x+5)(2x+3) = ...

 

Yes, there is an infinite number of ways to write it down. I suppose the (x+2)(2x+3) is what your teacher wanted since it seems to be the easiest (looks nicest to me).

 

For the squares, there´s three solutions:

1) Most people will understand if you write x^2.

2) My keyboard has a button for it: x². Not sure if the button exists on US keyboards.

3) It´s a good idea to get used to TeX early which this forum is a good place for. For using TeX, you have to put the TeX code between [ math] and [ /math] (without the space). The command for putting something in the exponent is the "^" of 1) (one reason why most ppl will understand it), so in your case [math] 2x^2 + 13x +15 [/math] (hover the cursor over the image to see the code I entered).

[the tree]

Factor generally means integer factor. If you consider numbers to be composites of other numbers, rather than merely different sized blocks then perhaps it'll be clearer why we don't really care for every single real divisor that a number has.

[ajb]

Isn't this related to the notion of irreducible polynomials and Galois theory?

[Dave]

Yes it is; the field is very interesting. I took a course on Algebraic Number Theory last year (which has a certain amount of overlap with Galois Theory). Studying the reducibility of polynomials over Q is a little dull; there's a much richer theory behind the whole lot if we only consider polynomials which are (ir)reducible over Z.