THERE ARE SOME SPECIAL FACTORIZATION –
Difference of two Squares –
We know that - (a²- b²) = (a + b) (a – b)
Example.1) Factorize => 16x²- 9y²
Ans.) 16x² - 9y²
= (4x)² - (3y)²
= (4x + 3y) (4x – 3y) (Ans.)
Example.2) Factorize => 1 – (x – y)²
Ans.) 1 – (x – y)²
= 1² - (x – y)²
= (1 + x – y) (1 – x + y) (Ans.)
Example.3) Factorize => 5aᶟ - 125a
Ans.) 5aᶟ - 125a
= 5a (a² - 25)
= 5a (a² - 5²)
= 5a (a + 5) (a – 5) (Ans.)
Example.4) Factorize => xᶟ - 3x² - x + 3
Ans.) xᶟ - 3x² - x + 3
= x (x² - 1) – 3 (x² - 1)
= (x – 3) (x² - 1)
= (x – 3) (x – 1) (x + 1) (Ans.)
Example.5) Factorize => 25x²- y² + 12 yz – 36z²
Ans.) 25x² - y² + 12 yz – 36z²
= (5x)²- {(y)²- 2.y.6z + (6z)²}
= (5x)²- (y – 6z)²
= {5x + (y – 6z)} {5x – (y – 6z)}
= (5x + y – 6z) (5x – y + 6z) (Ans.)
Sum or Difference of Cubes –
1) (aᶟ + bᶟ) = (a + b) (a² - ab + b²)
2) (aᶟ - bᶟ) = (a – b) (a² + ab + b²)
Example.1) 24a⁴ + 81a
Ans.) 24a⁴ + 81a
= 3a (8aᶟ + 27)
= 3a {(2a)ᶟ + (3)ᶟ}
[ applying the formulae (aᶟ + bᶟ) = (a + b) (a² - ab + b²) ]
= 3a (2a + 3) (4a² - 6a + 9) (Ans.)
27
Example.2) 64xᶟ - -------
xᶟ
27 3
Ans.) 64xᶟ - ------- => (4x)ᶟ - (-----)ᶟ
xᶟ x
3 3 3
= (4x - ------) {(4x)² + 4x . ------ + (-----)²}
x x x
[ applying the formulae (aᶟ - bᶟ) = (a - b) (a² + ab + b²) ]
3 9
= (4x - ------) (16x² + 12 + ------) (Ans.)
x x²