# CLASS-9FACTORIZATION BY TAKING OUT COMMON FACTOR

FACTORIZATION

Factorization is one of the most essential and important subject for 9th-grade math. The process of writing an expression as the product of two or more expressions is called factorization. Factorization is nothing but a reverse process of multiplication. Each expression occurring in the product is called a factor of the given expression.

Example.1) Factorize (x²- 2x – 3)

We can write, (x²- 2x – 3) = x²- (3 – 1)x – 3

=  x² - 3x + x – 3

=  x (x – 3) + 1 (x – 3)

=  (x – 3) (x – 1)

So, (x – 3) and (x – 1) are factors of (x²- 2x – 3)

In this book, we shall deal only with some special types of expressions for factorization.

Factorization by taking out the common factors

When each term of an expression has a common factor, we divide each term by this factor and take it out as a multiple, as shown below.-

Example.1) Factorize =>  24x²- 8x

24x²- 8x

=  8x (3x – 1)

Here, 8x and (3x – 1) are factors of (24x²- 8x) and 8x is the common factor.        (Ans.)

Example.2) Factorize =>  3ax + 3ay – 5bx – 5by

3ax + 3ay – 5bx – 5by

=   3a (x + y) – 5b (x + y)

=   (x + y) (3a – 5b)

Here, (x + y) and (3a – 5b) are factors of 3ax + 3ay – 5bx – 5by and (x + y) is a common factor.    (Ans.)

Example.3)  Factorize =>  45x²- 60xy + 20y²- 18x + 12y

=  5 (9x²– 12xy + 4y²) - 6 (3x – 2y)

=  5 (3x – 2y)²- 6 (3x – 2y)

=  (3x – 2y) {5(3x – 2y) – 6}

= (3x – 2y) (15x – 10y – 6)

Here, (3x - 2y) and (15x – 10y – 6) are factors of 45x² - 60xy + 20y² - 18x + 12y and (3x - 2y) is a common factor.              (Ans.)

Example.4) Factorize => x⁴ - 6xᶟy + 6x²y² - 8xyᶟ + 3xᶟy² - 6x²yᶟ

x⁴ - 6xᶟy + 6x²y² - 8xyᶟ + 3xᶟy² - 6x²yᶟ

=  x (xᶟ - 6x²y + 6xy² - 8yᶟ) + 3x²y² (x – 2y)

=  x (x - 2y)ᶟ + 3x²y² (x – 2y)

=  (x – 2y) {x (x - 2y)² + 3 x²y²}

=  (x – 2y) { x (x² - 4xy + 4y²) + 3x²y²}

=  (x – 2y) (xᶟ - 4x²y + 4xy²+ 3x²y²)

Here, (x - 2y) and (xᶟ - 4x²y + 4xy² + 3x²y²) are factors of x⁴ - 6xᶟy + 6x²y²- 8xyᶟ + 3xᶟy²- 6x²yᶟ and (x - 2y) is a common factor.   (Ans.)