FACTORING is a major portion of the math classes required after Algebra 2.

I will begin each of these lessons with easy problems from Algebra 1.  But by the end of each lesson you will be working problems that require the same concepts, but that are at an Algebra 2 level.   

Be sure you read and complete all examples in these lessons.   WRITE them into your notes.      You will need them for other Lessons in this course, its sequel and in future algebra or precalculus or calculus courses.

Looking back at 5x²y - 6xy, a monomial we could have used was

         x times the binomial 5xy- 6y

        or the monomial y times the binomial 5x²- 6x.  Both of the pairs multiply back to give the 5x²y - 6xy.

But we must choose to use  xy , that is the greatest common factor,

          ( x and y are just common factors).  

Remember we will always use the greatest common factor of all the terms.

For 10w²- 5w³+15w  we use 5w, the greatest common factor.  

                 5 alone or w alone would be just common factors.

    We must use the GREATEST!

$$$  Answers for the above practice problems are: xy times  5x - 6

          and 5w times (2w - w²+ 3).

Example:  42w²x - 6wx +36x .

  1.  There are three terms. 

  2.   Look at the three coefficients:  42 and -6 and +36.   

        They all have in common what number factor?  6    

  3.   NOW look at the three terms.    

       They all have in common what variable factor?  x         

              and What is the lowest value of the exponent of  x?  1 

  4.  So the greatest common factor for all three terms is 6x. 

Factor out the 6x from the three terms:  

      Divide (opposite of multiply) all the terms by the 6x to

           find the 3 new terms inside the parenthesis.   

Please look closely at the exponents and variables.   Recall x divided by x = 1.

42w²x - 6wx +36x   Divide each term by the 6x.

Divide the 42w²x BY   6x    

   and the - 6wx  BY 6x

    and the +36x BY    6x. 

                            We get 7w²- w + 6  .

          Answer is    42w²x - 6wx +36x   =    6x(7w²- w + 6)

Multiply the three terms inside the parenthesis by the 6x to check your answer.

Example :  The greatest common factor is   7x²y  

        for   14x⁴y - 42x³y- 63x²y  .

CAN YOU show that 14x⁴y - 42x³y-63x²y  =  7x²y( 2x²- 6x- 9)   ?

The steps for finding the 7x²y :

1.  There are 3 terms to consider. 

2.  They all have in common what number factor?  7   

3.  They all have in common what variable factor?  x and also y         

       and What is the lowest value of the exponent of x?    2

       and What is the lowest value of the exponent of y?    1

4.  So the greatest common factor for all three terms is 7x²y. 

Factor out the 7x²y from the three terms. 

Divide (opposite of multiply) all the terms by the 7x²y  to find the 3 new terms inside the parenthesis.    

Please look closely at the exponents and variables.   

Hope you can see the colors here.

Divide the 14x⁴y by 7x²y 

AND the - 42x³y by 7x²y 

 AND the -63x²y by 7x²y    =         2x²- 6x - 9

 That is

             14x⁴y - 42x³y - 63x²y  =    7x²y ( 2x²- 6x- 9)   

*************************************************************************

Be sure you understand what is going on with the exponents before going to the next example;  that is, when do we add them and when do we subtract them, and when do we multiply them. 

Do you know what  x x equals?  

What about x^2a x^a  ? 

Each of these involve multiplying,

   there is a common base  x ,

  and each has an exponent.  

Do you know how to simplify  x³x² ?   

is this x⁵or x^6 .     Where are your notes from Lesson 2.2?       

              x³x^{2   =} x⁵ 

We are multiplying so we can add the exponents!! = x^5 .

BUT NOW LOOK:

How do you simplify (x³) ² ? 

the  ²    means write the x³down twice. (x³) ^2  =(x³)(x³) 

We now have a multiplying problem so we add the exponents =x^{6 .}   

Or you can recall:

when an exponent is "on an exponent" then

       we can multiply the exponents.   

(x³)² = x⁶

Who can get this one right? 

**Does  x^5a= x⁵x^a or does  x^5a= x^3ax ^2a ?   

Add the exponents to find out.  

** which one above equals 5a?    

          IS it 5+a or is it  3a+2a?   

          I hope you know 5+a is adding UNLIKE terms

               and we can NOT add these unlike terms.

Answer:   x^5a= x^3ax^2a  but also we know: x^5a= x^4ax^a.

A common factor for X^3a- X^a+1+  4X^a  is  X^a.  Use the steps from above.

1.   There are 3 terms to consider. 

2.  They all have in common what number factor?     only a 1   

3.  They all have in common what variable factor?          X        

     and  What is t he lowest valueof the exponent of  X?    

    We will say 1a, but that really is tricky since we do not know its value.  

    Most algebra books will use the 1a.

4.  So the greatest common factor for all three terms is 1X^a.  

     Factor out the 1X^a from the three terms.

Divide the three terms by X^a.   (that is subtract the a from the exponents)

X^3a- X^a+1 + 4X^a  =

          X^a  ( X^{3a - a} - X^{a+1- a}  + 4X^{a- a} ) =     

HOPE you see that 3a - a = 2a NOT 3. !!! 

The 3a - a is same as 3a - 1a which = 2a.

          X^a(X^2a-X¹+4X⁰) =  Recall x⁰ is 1.

           X^a(X^2a-  X¹+ 4 ).

To check, multiply the 3 terms inside the parenthesis by X^a.

To multiply we add the exponents.