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 x^{2} + 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?

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.

Remember distributive property? It is

a (b + c ) = ab + ac

**Example 1: Factorize ax + bx**

Solution:

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.

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?
**

*Solution:*

12x^{2} + 18x^{3} can be factorized into any of the following ways:

12x^{2} + 18x^{3} = x (12x +18x^{2})

12x^{2} + 18x^{3} = 6x (2x + 3x^{2})

12x^{2} + 18 x^{3} = 6x^{2} (2 + 3x)

There may be more ways of expressing the polynomial 12x^{2} + 18x^{3} 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, 6x^{2}is the __ greatest common divisor__ or

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

**Example 1: Factorize x ^{2}y^{2} + y^{2}**

*Solution:*

Completely factorize x^{2}y^{2} + y^{2}, i.e. write the greatest common factor of x^{2}y^{2} and y^{2} as the common factor.

y^{2} is the highest common factor. To find the other factor, divide each term by the H.C.F. and add the quotients.
(x^{2}y^{2})/y^{2} = x^{2} and y^{2}/y^{2} = 1

Sum of the quotients is x^{2} + 1, which is the other factor. Therefore,

x^{2}y^{2} + y^{2} = y^{2}(x^{2} + 1)

**1. Factorize a ^{2} + bc + ab + ac**

let us group terms in a

But, which terms to group?

Those terms which yield a common factor on grouping!

Group the terms like this:

a^{2} + ab + bc + ac

a is the common factor in a^{2} + ab and b is the common factor in ab + bc

a^{2} + 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

a^{2} + ab + bc + ac =

a (a + b) + c (a + b) =

(a + b) (a + c)

(a + b) and (a + c) are two binomial factors into which a^{2} + ab + bc + ac
is factorized.

2. Factorize ax + bx + ay + by

*Solution: *

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 p^{2} qx + pq^{2}y + px^{2} y+ qxy^{2}

*Solution:*

Group p^{2}qx and px^{2}y.

Common factor is px

Factorization of p^{2}qx and px^{2}y = px (pq + xy)

Again, group pq^{2}y and qx y^{2}

common factor is qy

Factorization of pq^{2}y and qxy^{2} = qy (pq + xy)

So we have :

p^{2} qx + pq^{2}y + px^{2} y+ qxy^{2} = px (pq + xy) + qy (pq + xy)

In the next factorization step,

pq + xy is the common binomial factor. We have:

(pq + xy)(px + qy)

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

Example

a^{2} + 2ab + b^{2}

In this trinomial, a^{2} and b^{2} are the two perfect squares and 2ab is the product of the square roots of a^{2} and b^{2}
Can you do the Factorization?

Factorization ofa^{2} + 2ab + b^{2} gives the famous formula:

(a + b) ^{2} = a^{2} + 2ab + b^{2}

**Example 1:
Factorize 9p ^{2}+ 24pq + 16q^{2}**

9p

16q

24pq = 2(3p)(4q), just like 2ab

Applying, (a + b)

Factorization of

9p

**Example 2:
Factorize -4x ^{2} + 12x + 9**

In 4x

Whenever the leading coefficient is negative, express the given polynomial as follows:

-1(4x

Now, use factorization of perfect trinomial squares method to factorize

4x

a

so, 4x

Now, factorization of

4x