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Factoring

Common Factoring

First thing you should ALWAYS try to do is to common factor if possible.  Well what is common factoring? 

 

Pretty easy you just commom factor out a number if there is one for example,

y=4x+6  

In this equation every number can be divided by 2 therefore you can common factor out a 2.

y=2(2x+3)  Pretty simple right? 

Video on  Factoring a Simple Trinomial!

Video on Factoring  a  Complex Trinomial!

Difference of Squares

What is a difference of squares? 

A difference of squares is where 2 terms are perfect squares.   They have to be a difference of squares NOT a sum of squares.  For example:

4x²-16       and NOT    4x²+16

 

The formula for difference of squares is:

(a+b)(a-b)=a²-b²

 

So the solve for 4x²-16 first you would common factor out a 4

 

4x²=16

=4(x²-4)

a=√x²=x  Do the root of x² and 4 (Note that 

b=√4=2  the negative sign is not related

                to 4 as a root of a negative number

               is not possible

=4(x+2)(x-2)  Bring the 4 that you factored                           out first and use the x and 2                             you got as it is in the formula                           (a+b)(a-b) like (x+2) (x-2)

Video on  Difference of Squares and Perfect Square Trinomials!

Simple Trinomial Factoring

What are simple trinomials?

Simple trinomials are expressions that are trinomials (which means 3 terms) with the first term ax² having "a" equal to 1.  

 

When factoring a simple trinomial such as 

x²+7x+12 you always want to find:

2 numbers that ADD to give b

2 numbers that MULTIPLY to give c

 

If you don't know what I'm talking about then look at the standard form formula which you factor from: ax²+bx+c  

 

For example I will show you how factor 

x²+7x+12

=(x+3)(x+4)     As you can see 3 and 4 add up                        to give b which is 7 and they                            multiply to give c which is 12                           (yes this does require some                             thinking).

Complex Trinomial Factoring

What are complex trinomials?

Sound complex? Don't worry once you know simple trinomials complex are pretty easy.  Complex trinomials have a coefficient other then 1 in front of the x² term.  

 

Before you can factor complex trinomials you need to know 2 things: 

=Binomial Common Factoring

=Factor By Grouping

Binomial Common Factoring:

3x(z-2)+2(z-2)  Bascially you take the term 

=(3x+2)(z-2)       outside of the bracket and                                put them together then take

                           the SAME (has to be same)                                terms inside the brackets                                 and put them together.

Factor By Grouping:

df+ef+dg+eg           |NO common factors|

=(df+ef)+(dg+eg)      Basically you 

=f(d+e)+g(d+e)          make brackets                                                     around 2 terms                                                   that have                                                             something in                                                       common and                                                       factor them                                                         out. 

Factor the following trinomial using the Decomposition Method (most commonly used to factor complex trinomials):
6x²+7x-3    First find the product of term a     6x(-3)=-18   and term c 

(9)x(-2)=-18 After doing the first step you do

(9)+(-2)=7     the same as you did in simple                          trinomials and find two numbers                      that multiply to give -18 and                            two numbers that add up to                            give 7

=6x²+9x=2x-3  Now replace the middle term                          with the two numbers (and x                            because the middle term also                          has x) 

=(6x²-2x)+(9x-3) Now just group them

=2x(3x-1)+3(3x-1)  If done correctly the                                         terms in the brackets                                         should be the same

=(2x+3)(3x-1)    Just binomial factoring

Perfect Square Trinomials

What are prefect square trinomials?

The trinomial that results from squaring a binomial is called a PERFECT SQAURE TRINOMIAL.

 

The formula for a perfect sqaure trinomial is:

a²+2ab+b²=(a+b)²    or (depending on the equation) a²-2ab+b²=(a-b)²

 

The first and last terms MUST be positive and perfect squares.

 

9a²-48a+64

This is a perfect square trinomial, well how do we know? 

 

You take the first and last term and square root them because the middle term of the equation is 2ab and the first and last terms are squared (a² and b²).

√9a²=3a

√64=8

The middle term is 2ab so let's subtitute the values:

-2(3a)(8)=-48a 
The a and b values are multiplied by -2 because the middle term of the original equation is negative and if the answer is equal to the middle term the equation is a perfect square trinomial.

 

Well how do we factor it though?

9a²-48a+64

√9a²=3a          Basically do the same thing and 

√64=8             take the roots of the first and                               last terms.

=(3a-8)²       Now just write the 2 numbers you                       get and put the sign of the middle                       term which in this case is -48a so                       the sign is negative and put a                               square outside of the brackets                           that you would put the 2 terms in.

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