## Sunday, September 20, 2015

### Factorization

Factorization is the easiest thing. It is really easy to mess up but it is also easy to do when you know all the rules and how to apply them (see what I did there).

The Number 1 Rule of Factorization (Common Factors)

The first thing we do when we factorize is look for common factors.

$$7x^2 - 3x$$

So to factorize this we just look for a common factor (x in this case) and we cannot go any further

$$x(7x-3)$$

Difference of 2 Squares

$$x^2 - y^2 = (x-y)(x+y)$$

So we take the square root of each square and +, - but why is that? Is it just a magical formula I must learn? Of course not.

$$(x-y)(x+y) = x^2 + xy - xy - y^2$$

See if we expand this then we always + and - the same middle term so it disappears and this is where the concept of the difference of two squares comes from.

E.g.

$$100x^3y - xa^2y$$

You might look at this and be confused but remember what the number 1 rule is?

$$=xy(100x^2 - a^2)$$

Ah we can take out a common factor of xy.

Now:

$$=xy((10x)^2-(a)^2) = xy(10x-a)(10x+a)$$

If there are only 2 terms for example then look at the first example I gave at the top of this post.

For things that look like this:

$$x^2+6x+5$$

So we look at the last sign. It's positive. Oh so if we have two brackets (_+_)(_+_) or (_-_)(_-_) then these are the only two that will make a positive number for that 5.

$$x^2+6x+5$$

Now look at the first sign. It's also positive. The terms that we add are either both positive or both negative so in this case we will choose both to be positive since there is a +6

$$=(x+5)(x+1)$$

Note that the last two numbers have to make the 5 at the end when multiplied together

$$x^2- 6x +5$$

Oh so in this case we say

$$=(x-1)(x-5)$$

$$2x^2-25x-13$$

The negative here means that we have - and +.

The 2 has to go with one of the x's

$$(2x +1 )(x -13)$$

The more you practice this the better you will get but you need to think about where to put things and you can multiply out at the end to see if your answer is right.

For this one above we have a negative first sign so we know that the larger number needs to be the - one and since we are looking for negative 25 it makes sense that -2(13)+1 = -25.

Four Terms?

$$-x^2 - 2x - x -2$$

Try to factorize two terms and see if we can take out a common factor.

$$-x(x+2) -1(x+2)$$

$$=(x+2)(-x-1)$$

$$-(x+2)(x+1)$$

See how I took (x+2) out.

These are the basics of factorization. There are some more advanced forms of factorization but you need these as a foundation.