3:5 Factoring Completely


LESSON 3.5 Factor Completely

 

Some factoring problems require several steps to be factored completely.

We have studied several types of factoring.  Let's review these. 

 

You should write an example or two for each type because in this lesson and throughout the course we will use these and they will be mixed together.  

 

  These are the 6 categories of factoring we have studied.  

In this lesson we will use more than one of these in each problem that we have.  

 

 

BEFORE we begin this LESSON I want you to review.    

Here is a link to a game that I created.

You will factor polynomials that have 2 or 3 terms.      

This will give you easy practice in the CATEGORIES A., C., D., and E. below.       

 

Factoring Polynomials (3 or less terms)http://www.quia.com/cb/369639.html       

Let me know your score by emailing it to me.  

You should play the game more than once.  

You may start over any time too.  

Type the answers as we have been doing in this class.  

BUT DO NOT USE the SPACE bar in the answers. 

 

 

 

 NOW  YOU  will learn to COMBINE THE  types of "factorings" !!!

 

 

The CATEGORIES of FACTORING :

A.  (Greatest) Common factoring:  7x2+14xy    = 7x(x +2y)   Lesson 3.1

B.  Grouping(4 terms usually):   Lesson 3.2  Big P principle

      4x-8y +2xm-4ym =

      4(x-2y)+2m(x -2y)=

      4P+2mP     =        2P[2 +m]     =     2(x-2y) [2+m]

C.  Quadratic Trinomials:  3x2+5x - 2 = (3x -1)(x +2) Lesson 3.3

D.  Cubic binomials:  x3+ 8    or     64y3 - x6      Lesson 3.4 .                     

Recall that gave us a short parenthesis times a long parenthesis!!  

E.  Differences of Squares:   x2- 9 = (x-3)(x+3)         Lesson 3.4B

or  look at this "double" difference of squares:                               

y4- 81     = (y2- 9)(y2 + 9) , but we have another diff. of squares (see pink) 

                  we get (y-3)(y+3)( y2 + 9) 

F. More Big P Principle:  (x+4)2-9    becomes  P2  - 9  Hope you can factor that.

                           And this one too (x-2)2+ 2(x-2)+1   becomes  P2+ 2P +1

and then we use category E or C to finish.    

 

The Big P Principle can be used with any of the other methods.

Don't  forget the  P is temporary. P is temporarily replacing the (parenthesis value).

 

Big HINT: 

 

Always, Always, Always look for and do greatest common factoring, CATEGORY A above, FIRST if the problem contains a common factor.  Then look for any of the other Categories.

 

****These problems below take at least 2 steps of factoring.  

Do not try to do them in one step.  

It is rarely possible for anyone to factor these correctly if they try to do them in one step.***************

 

1. 20x2 - 38x3- 30x4 What do you see that is in common to all three terms?   

2. 6y3+ 11y2- 35y

 

3. 18x2 - 200y2

4. 64y4- 125y

 

5. x4+ 5x2+ 6  Hint:  Use the Quadratic Box and the video from Lesson 3.3   

                     but under the F write x2 and x2 rather than our usual x and x .

6. (y-2)2- 9 think P2- 9   

now use category E and then replace the P with the (y- 2).

 

7. (y+5)4- 7(y+5)2 + 12           

                                      Think P4-7P2+12    

8.  81m4 - y4

 

COPY the answers below into your notes.           

This is not optional!!

Find answers by ^^^^^^^

 

 

 

 

^^^^^^^ 

1. 20x2- 38x3- 30x4 Category: (A) then (C) above

=2x2(10 - 19x - 15x2) Category A

=2x2(2 -5x)(5 +3x) Category C

 

2. 6y3+ 11y2- 35y   

category: (A) then (C) Do you see the common factor first?

= y(6y2+11y -35)

=y(3y -5)(2y +7)

 

3. 18x2 - 200y2 =

2(9x2 - 100y2) = Category A

=2(3x-10y)( 3x+ 10y)   Category E  differences of squares.

 

4. 64y4-125y    category A then  category D.   

Do you see the common factor of y?

= y(64y3-125) see the cubics

= y (4y -5)(16y2+ 20y +25)

 

 

5.   x4+ 5x2+ 6  = (x2+ 2)(x2+ 3) category C

 

6.   (y-2)2- 9                            

think P2- 9      

Use Big P then category E.

=(y - 2 - 3)( y - 2 + 3) = and that simplifies to (y-5)(y+1)  .

 

7. (y+5)4- 7(y+5)2 + 12               

Let the y+5 be BIG P, briefly.  

Think   P4-   7P2   + 12                   

Factor the quadratic trinomial ( P2____)( P2_____)

DID you get

= [(y+5)2-3] [(y+5)2-4] 

Use Big P then category C.

 

8.  81m4   - y4

=(9m2-y2)(9m2 + y2)    

but see the difference of square in first parenthesis.

=(3m-y)(3m+y)(9m2 + y2).

 

 

 Do Assignment 3.5 in the Navigator Box

Did you learn to do this problem earlier?   It is only one category!

 

Only for the BRAVE:

 

Can you common factor when the exponent is a variable too!

 

x2n- xn+1+ xn     do you see all have xn  in common. 

 

Factor xn  out!    

 

BUT first do you recall that when multiplying we add the exponents.  

 

Thus xn  times  xn    is x2n   ( I added n+n).    

 

So what is xn  times x.    What are the exponents? 

n and 1  so we get   xn+1. 

 
 

x2n- xn+1+ xn        You may want to think xnxn - xnx1+ 1xn .

 

factors to xn(xn- x1+ 1)        

                   OR  xn(xn- x + 1)