Factoring formulas
Factorization, also known as Factoring, is a process of breaking down a large number into several small numbers. When these small numbers are multiplied, we will get the actual or original number.
1. a2 – b2 = (a + b)(a – b)
Example:
Factor: x2 - 4 y2
Solution:
x2 - 4 y2 = x2 - (2 y)2
x2 - 4 y2 = (x - 2 y)(x + 2 y)
2. a3 – b3 = (a – b)(a2 + ab + b2)
Example:
Factor: 8x3 - 27 y3
Solution:
8x3 - 27y3 = (2x)3 - (3 y)3
= (2x - 3y)[(2x)2 + (2x)(3 y) + (3 y)2 ]
= (2x - 3 y)(4x2 + 6xy + 9y2 )
3. a3 + b3 = (a + b)(a2 – ab + b2)
Example:
Factor: 54x3 + 16 y3
Solution:
54x3 + 16 y3 = 2(27x3 + 8 y3 )
= 2[(3x)3 + (2 y)3 ]
= 2(3x + 2 y)[(3x)2 - (3x)(2 y) + (2 y)2 ]
= 2(3x + 2 y)(9x2 - 6xy + 4 y2)
4. a4 – b4 = (a – b)(a + b)(a2 + b2)
Example:
Factor: 48x4 - 3y4
Solution:
48x4 - 3y4 = 3(16x4 - y4 )
= 3[(4x2)2 - ( y2)2]
= 3(4x2 - y2)(4x2 + y2 )
= 3(2x - y)(2x + y)(4x2 + y 2 )
5.a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4 )
6. a5 + b5 = ( a + b)(a4 - a3b + a2b2 - ab3 + b4)
7. If n is odd, then an + bn = ( a + b)(an-1 – an-2b + an-3b2 - … - abn-2 + bn-1).
8. If n is even, then an – bn = ( a - b)(an-1 + an-2b + an-3b2 + … + abn-2 + bn-1). an + bn = ( a + b)(an-1 – an-2b + an-3b2 - … + abn-2 - bn-1).
