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Algebraic fractions and factorising quadratics
By Patrick Todd on November 9, 1998:

At school I have started GCSE and at the moment we are using quadratic equations in algebraic fractions, I am finding it hard to find the right set of numbers and the correct order of the plus, minus. Can you help with any types of methods to use to make it easier.

e.g.


 1
2m2+3m-2
 -  1
3m2+7m+2
=
 1
(2m-1)(m+2)
 -  1
(3m+1)(m+2)
=
 1(3m+1)-1(2m-1)
(m+2)(2m-1)(3m+1)
=
 3m+1-2m+1
(m+2)(2m-1)(3m+1)
=
 m+2
(m+2)(2m-1)(3m+1)
=
 1
(2m-1)(3m+1)
 (cancelling the (m+2))


What I would like to know is an easier way of finding the second line(the first pair of brackets, quadratic).

thanks alot
Patrick Todd
By Gordon Lee on November 24, 1998:

I am afraid there aren't many ways of making it easier, since what you do are still very basic operations. There isn't really many 'clever tricks' you can do...

Personally, I hate algebra more than anybody else, since it is all very messy and I usually get it WRONG! Mathematics is much more than just doing algebra and I usually just cheat and shove it in a computer.

Sorry that I can't be of much help,

Gordon

By Richard Dwight on November 26, 1998:

As far as I can see, your problem is how to factorize quadratic equations. This is a classic pet-hate at GCSE level, but once you've done a few (hundred) times they get trivial.

I can give you a couple of algorithms for factorizing quadratic equations, but they are inherently messy, and you will soon find that it becomes much easier to do it by eye. Hopefully these will tide you through for now.

If you understand what the factorization of a polynomial means, you can see how to come up with a general rule for finding it. To see what it means, think of a specific case:
e.g. x2+5x+6=(x+3)(x+2)
The numbers in the factorization of a quadratic are exactly the root a of the quadratic. (A root of a function is where the graph of the function crosses the x-axis, or a number x, such that f(x)=0.) So in the case above f(-2)=4-10+6=0 and f(-3)=9-15+6=0. This is obvious if you look at the right hand side above, x=-2 takes x+2 to zero and so the whole right hand side to zero. Similarly for x=-3.

So you can see that if you can find the roots of the quadratic you can factorize it, with the factors just (x-{the roots}).

If you know how to find the roots of a quadratic, you can now factorize them. A couple of methods of finding roots are completeing the square, and using the formula:

root =
-b ±
Ö
b2-4ac
2a
If you don't know these already then you'll come across them soon, or if you're really impatient you can write back.

It's interesting to note that the method of finding the factorization by finding the roots works for all polynomials, however it's a difficult problem to find the roots of a general nth degree polynomial, and so difficult to factorize a general nth order polynomial. You won't need to worry about this for a while yet though.

I hope this helps some,
Write again soon,
Richard