Basic Algebra > Factorization

What is factorization?

Consider the following example:
12 = 3 × 4
i.e., 12 is product of 3 and 4.
3 and 4 are called factors or divisors of 12
12 is also equal to 2 × 6.

Here again, numbers 2 and 6 are called factors or divisors of 12.
The process of writing number 12 into product of 3 and 4 or 2 and 6 is called factorization.

Which is the other way of factorizing 12?
It is 12 = 12 × 1
Above is an example of factorizing a monomial (here 12) into three pairs of other monomials 3, 4 and 2, 6 and 1,12.

An algebraic expression can also be factored into one or more factors.
For example, the polynomial x2 + 2x can be written as product of two binomials namely, x and (x+2)
x and (x+2) are called factors or divisors of x and (x+2) .
Also, they are polynomials, while, x is monomial and x+2 is a binomial.

So, what is Factorization?


Expressing polynomials as product of other polynomials that cannot be further factorized is called Factorization

We will discuss all the various types of factorization in this lesson.
They are standard types found across all books on algebra. Here, we go.

1. First Type: Factorization into Monomials:

Remember distributive property? It is
a (b + c ) = ab + ac

Example 1: Factorize ax + bx

In ax and bx, x is common and also a common factor.
Write the common factor x in the polynomial ax + bx outside as x(a + b)
Now, both x and a + b are factors of the polynomial
ax + bx
So, factorization of
ax + bx = x(a + b)
Here the common factor x is a monomial.

Important Note:

In 10(a + b), 10 is one of the two factors and we do not factorize 10. We need not write 10 as 1×10 or 2×5. Factors that are numbers are not further factorized.

What is Complete Factorization?

Factorize: 12x2 + 18x3

12x2 + 18x3 can be factorized into any of the following ways:
12x2 + 18x3 = x (12x +18x2)
12x2 + 18x3 = 6x (2x + 3x2)
12x2 + 18 x3 = 6x2 (2 + 3x)

There may be more ways of expressing the polynomial 12x2 + 18x3 as product of two factors. Let us limit to the above three ways.

Which of the above three ways is accepted as standard form of factorization?

The third one.
In the third one, 6x2is the greatest common divisor or highest common factor of the two algebraic expressions (monomials) 12x2 and 18x3

Complete Factorization

In complete factorization, the greatest common factor is written as the common factor

Example 1: Factorize x2y2 + y2

Completely factorize x2y2 + y2, i.e. write the greatest common factor of x2y2 and y2 as the common factor.
y2 is the highest common factor. To find the other factor, divide each term by the H.C.F. and add the quotients. (x2y2)/y2 = x2 and y2/y2 = 1
Sum of the quotients is x2 + 1, which is the other factor. Therefore,
x2y2 + y2 = y2(x2 + 1)

2. Second type of Factorization: By Grouping of Terms

1. Factorize a2 + bc + ab + ac
let us group terms in a2 + bc + ab + ac
But, which terms to group?

Those terms which yield a common factor on grouping!
Group the terms like this:
a2 + ab + bc + ac
a is the common factor in a2 + ab and b is the common factor in ab + bc
a2 + ab = a (a + b) and bc + ac = c (b + a)

now, the common factor is the binomial (a + b).

{note that a + b is same as b + a, i.e., order of addition does not matter, since addition follows closure property which states a + b = b + a}

So, factorization of
a2 + ab + bc + ac =
a (a + b) + c (a + b) =
(a + b) (a + c)
(a + b) and (a + c)
are two binomial factors into which a2 + ab + bc + ac is factorized.

2. Factorize ax + bx + ay + by

Group terms with similar literal coefficients and numerical coefficients.
One group is ax + bx, in which x is the same literal coefficient,
and the other is ay + by, in which the y is the same literal coefficient,
In ax + bx, the common factor is x and in ay +by, the common factor is y.
on factorization with common factors, we have:
ax + bx = x ( a + b ) and
ay + by = y (a + b)

so ax + bx + ay + by = x (a + b) + y (a + b).
(a + b) is the common binomial factor for the next step of factorization:
(a + b)(x + y)
Factorization of ax + bx + ay + by = (a + b)(x + y)

3. Factorize p2 qx + pq2y + px2 y+ qxy2

Group p2qx and px2y.
Common factor is px
Factorization of p2qx and px2y = px (pq + xy)
Again, group pq2y and qx y2
common factor is qy
Factorization of pq2y and qxy2 = qy (pq + xy)
So we have :
p2 qx + pq2y + px2 y+ qxy2 = px (pq + xy) + qy (pq + xy)
In the next factorization step,
pq + xy is the common binomial factor. We have:
(pq + xy)(px + qy)

Type 3: Factorization of Trinomial Perfect Squares

Trinomial Perfect Squares have three monomials, in which two terms are perfect squares and one term is the product of the square roots of the two terms which are perfect squares

a2 + 2ab + b2
In this trinomial, a2 and b2 are the two perfect squares and 2ab is the product of the square roots of a2 and b2 Can you do the Factorization?

Factorization ofa2 + 2ab + b2 gives the famous formula:
(a + b) 2 = a2 + 2ab + b2

Example 1:
Factorize 9p2+ 24pq + 16q2

9p2 = (3p)2, just like a2
16q2 = (4q)2, just like b2
24pq = 2(3p)(4q), just like 2ab
Applying, (a + b) 2 = a2 + 2ab + b2
Factorization of
9p2 + 24pq + 16q2 = (3p + 4q)2 = (3p + 4q)(3p + 4q)

Example 2:
Factorize -4x2 + 12x + 9

In 4x2 + 12x – 9, the leading coefficient is –1.
Whenever the leading coefficient is negative, express the given polynomial as follows:
-1(4x2 – 12x + 9).
Now, use factorization of perfect trinomial squares method to factorize
4x2 – 12x + 9, which is in the form of the well-known algebraic formula:
a2 – 2 ab + b2 = (a – b) (a – b).
so, 4x2 – 12x + 9 = (2x – 3) (2x – 3)
Now, factorization of
4x2 + 12x – 9 = -1(4x2 – 12x + 9) = -1(2x – 3) (2x – 3)