Dictionary for Basic and Intermediate Algebra
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How To Factor An Expression

Factoring an algebraic expression is essential in mathematics. In arithmetic, you learn the fundamental theorem of arithmetic, it's about factoring composite numbers. In algebra, you learn the fundamental theorem of algebra, it's about factoring reducible polynomials. Here you will learn the basic techniques in factoring algebraic expressions.

(A)  Factoring a monomial from a polynomial.

Factor out (if there is one), the greatest common  factor of all terms in the expression

Ex 1:  Factor


     

Ex 2 :   Factor .

   

Ex 3:  Factor


 

Ex 4 :   Factor

(B) Factoring binomials of the following types.

(i)  Using the formula to factor the difference of two squares:

Ex 1:  Factor


Ex 2 : Factor .

Ex 3 :    Factor .
         

Ex 4 :   Factor .

(ii)  Using the formula to factor the sum or the difference of two cubes:
                 

Ex 1:  Factor
        

Ex 2:   Factor
                   

Ex 3:   Factor
         

Ex 4: : Factor .
        

Ex 5:   Factor
         
 

(C)  Factoring second degree trinomials:

(i) Perfect square of a trinomial:
(ii) Trinomial of the form
.

(i) Using the formula:

Ex 1: Factor

        
         

Ex 2 :   Factor 

Ex 3 :   Factor  
       
 
(ii)      Second degree trinomial of the form

Case 1:     when a = 1,

     

Ex 1:   Factor  
 

We look for two whole numbers whose product is 6 and their sum is 5
Decomposing 6 as product of two factors we get:

    
The desired factors are 2 and 3.
so

Ex 2 : Factor 
 
    
The desired factors are -16 and 3.
so  

Ex 3:   Factor 

None of the factors added up to be a sum of -7 and so the trinomial can't be factored, it is an irreducible polynomial.

Case 2: when a is not 1:

If the trinomial can be broken in to factor then , where p,q are factors of a and r,s are factors of c and bx is the sum of psx+qrx.

Ex 1:Factor

Factoring by grouping we get

Ex 2:Factor

Factoring by grouping we get

Ex 3: Factor

If factor then
Where p,q can be 15, 1 or 5, 3
and m,n can be 8, 1 or 4, 2

So

 

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