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 square**s:

**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