Algebra-- Factorization

This lesson will be focused on this mathematical concept called factorization.

Now, what is factorization?

Well, the 'wikipedia' definition of Factorization is : decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. 

This is correct, but too complicated. Now, lets explore my definition of factorization. Factorization is basically finding the HIGHEST common factors between a polynomial, numbers, etc and then taking the highest common factor out into a separate bracket so when the highest common factor of the polynomial, function, numbers is multiplied by the polynomial, function or numbers DIVIDED by the highest common factor, we get the original polynomial, function, numbers.

Now, this may sound complicated. Well, ill do a ton of examples to negate the effects of the tedious explanation.


Example 1 Factorize, (2x+4)


Now, to tackle this problem, we first break up the function into its individual components; there is +2x (positive 2x) and +4 (positive 4)


Now, we evaluate the two numbers in the function 2x and 4.

We need to find the HIGHEST COMMON FACTOR between the two numbers 2x and 4

Well, if you work this out through your basic algebra skills, you'll solve that 2x and 4 have a HCF of 2.

As you can see, the greatest number in which 2x and 4 both divides into is 2. If you have any problems with solving common factors (http://www.jamit.com.au/htmlFolder/FRAC1004.html) will be a great place to start!

Now, we find that the HCF is 2. 

The next step is to divide the HCF by the original function. Why? I''ll tell you later.

So if you divide 2x+4 by 2, you get +1x and +2.

Factorization has this rule, and the rule is that the overall answer should be in the form X(a+b), where X is the HCF factor and (a+b) is the function we receive after dividing the HCF by the original function, in this case 1x+2. 


Now, the sign X(a+b) is a shorthand notation stating that we have to multiply X by a+b. In this case, we have to multiply 2 by (1x+2)


If you do not know how to multiply out functions together, then I have another tutorial coming soon describing this process.


Anyway, if you multiply 2 by (1x+2), you get (2x+4)....hey..isn't this a bit like what I said at first?


Let me quote it..'.we get the original polynomial, function, numbers.'

And so did we...We got what we started with. So the process you just encountered is correct.
Multiplying the X by a+b is just a way to check that you have the correct answer. You end with the function at the beginning, you are correct. You dont, you must have made a mistake.


So the final answer is 2(x+2)


Note: 1x can be alternatively written as x.


Example 2:


Factorize 6x+9


Again, first, analyze the individual components, 6x and 9. Then, find the HCF between them. You should get 3. Then, divide 3 by 6x and 9 by 3. You should get 2x and 3


Awesome! now, lets put it into X(a+b) form.


X, the highest common factor is 3.
a+b will be 2x+3


So, the answer will be 3(2x+3)


and if you checked by multiplying the answer out, you should get what we started off with.