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.)