How do we divide polynomials?

Divide:  4x^12y^6

              36x^9y^7

When dividing polynomials ALWAYS remember the exponents get subtracted.  So first can you simplify 4 over 36 ?

                                                                                                                                    

Yes divide it both by 4 , so its 1 over 9 next you subtract the exponents . You have x^12 and on the bottom x^9 , 12-9=3 so its x^3 and then you see the Y variable , 6-7= -1 in other words just 1

FINAL ANSWER : 1 x^3

                         9 y                                                                                                                       

How do we factor the difference of squares ?

Factor:   x^2 - 9
Both x^2 and 9 are perfect squares. Since subtraction is occurring between these squares, this expression is the difference of two squares. This is where the term : Difference comes from.

What times itself will give x^2 ?  The answer is x. What times itself will give 9 ?  The answer is 3. In order words the square root of 9 is 3 .

The factors are (x + 3) and (x - 3).
Answer:  (x + 3) (x - 3)  or   (x - 3) (x + 3)  Order doesn't matter. 

How do we factor trinomials ?

 Factor x² + 14x + 24

So ,the best way to solve these problems is doing the diamond method. In other words the big X
The 24 on the bottom and on top of the X is 14. You need to find to numbers that multiply to 24 and add to 14. In this case its -> 2 and 12. 
Final answer should be (x+2)(x+12)

How do we solve diamond problems ?

To start of you draw a Big letter XOn the bottom is the product and on the top is the sum, but either way is correct. On either side are 2 numbers that when you multiply equals the number on the bottom/top and when you add equals the number on the top/bottom.

So . . . 

                
4+3 = 7 so that's good.


And . . 4 x 3 = 12 Check     

How do we multiply Polynomials?

When you multiply polynomials you simply use the distributive property. You multiply the numbers together and then you add the exponents with the same variables.

Example: -5y^2(4y^4+2y^2+1)

So . . -5y^2 * 4y^4 = -20y^6
         -5y^2 * 2y^2 = -10y^4
         -5y^2 * 1 = -5y^2

Then : -20y^6 - 10y^4 - 5y^2