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 
