x

(x + 2) (x + 3) =

x.x + x.3 + 2.x + 2.3 =

x

x

So, if you were asked to factorize x

First, understand that:

factorization of x

of the form (x +?) (x +?)…………(1),

where the two ? stand for some numbers.

So write 1 for numerical coefficients of x in each binomial factor.

Now, what to fill in? in (1) above.

Follow this Tip:

Which two numbers’ product is 6 and sum is 5?

You said it, didn’t you?

Yes, it is 2 and 3.

Therefore, write 2 and 3 in the? in (1) above.

So the two binomial factors are (x + 2) and (x + 3)

Therefore,

(x + 2) (x + 3) = x

Factorize

Leading coefficient of x

Set x

Where a and b are two numbers such that

a + b = 13, and a.b = 36

to find a and b, express 36 as product of pairs of its factors.

36 = 36 × 1,

36 = 2 × 18,

36 = 3 × 12,

36 = 4 × 9

from factorization of 36 into the above four forms, in the pair 4 and 9, the sum is 13, the numerical coefficient of the middle term in x

so, in the factorization of x

plug 4 in a and 9 in b {9 in a and 4 in b is equally correct}

x

ax

We learnt Factorization of x

How to factorize ax

Its almost the same, with only one slight change.

1. Think of two numbers whose product is a×c and

2. The sum of the two numbers must be b

Factorize: 3x

Observe that 3 is the numerical coefficient of the leading term. So it is of the form

ax

Compare the standard form ax

What do you find?

You find in the places ofa, b and c respectively 4, 13 and 9

a is 4, b is 13 and c is 9.

Now a.c =4.9 = 36

Think of two terms whose:

product is 36 (i.e., a.c) and

sum is 13 (i.e., b, the middle term)

factorize 36 into pairs of numbers as follows:

36 = 36×1,

36 = 2 × 18,

36 = 3 × 12,

36 = 4 × 9

Now, sum of which two factors is 13. They are 4 and 9

Write 4x

4x

Take out 4x as the common factor in 4x

and 9 as the common factor in 9x + 9 and factorize

4x

= 4x (x + 1) + 9 (x + 1)

= (x + 1) (4x + 9)

Factorize -6x

So, (-6x

Factorize (6x

on comparing the given polynomial with the standard form ax

a = 6, b = –1 and c = –1

Now,

a × c = 6 × (–1) = –6, and b = –1.

We need two numbers (factors) whose product is –6 and sum is –1

The numbers are 2 and 3. Put the ‘– ‘for the larger value 3 as sum is –1

{i.e. – 3 + 2 = - 1}.

Also 3 × (-2) = -6

6x

= 3x (2x –1) + 1(2x – 1)

= (3x + 1) (2x –1)

But (-6x

Factorize 12x

12x

Here a = 12, b = –1 and c = –1

Now, a × c = –12 and b = –1

4 × 3 = 12.

Adjust signs of 4 and 3 so that their sum is -1. So,

–4 + 3 = –1

Now,

12x

12x

4x (3x – 1) + 1(3x – 1) = (4x + 1)(3x – 1)

The polynomials (algebraic expressions) are two perfect squares separated by a **“negative sign”**.

Hence the algebraic expression is called ** "Difference of Two Perfect Squares"**.

And this type is called factorization of

Let one perfect square be:

a

then the difference of the two perfect squares looks:

a

you know very well that

a

In (a – b) (a + b):

a is the square root of a

So, to factorize:

• find square roots of each perfect square.

• In one factor, write their sum and in the other, write their difference.

Factorize 9x

9x

a

factorization of

9x

{order of factors does not matter, so it is equally correct to write the product as (3x + 4y) (3x – 4y)}

Factorize (p

first find square roots of the two perfect squares.

(p

Next, use the algebraic formula

a

to factorize(p

(p/2 – q/2) (p/2 + q/2)

Factorize 27y

3 is a common factor of 27 and48.

Write it outside the bracket as follows:

3 (9y

Now use the algebraic formula:

a

to factorize 9y

9y

9y

Therefore, 3 (9y

Factorize 4x

4x

4x

= (2x –5(y + z)) (2x + 5(y + z)) =

Now take the common factor 5 inside the bracket and write it as numerical coefficient for each of y and z

(2x – 5y – 5z) (2x + 5y + 5z)

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