22/09/2024
Algebra
Algebra is a branch of mathematics that deals with symbols and the arithmetic operations across these symbols. These symbols do not have any fixed values and are called variables. In our real life problems, we often see certain values that keep on changing. But there is a constant need to represent these changing values. Here in algebra, these values are often represented with symbols such as x, y, z, p, or q, and these symbols are called variables. Further, these symbols are manipulated through various arithmetic operations of addition, subtraction, multiplication, and division, with an objective to find the values. 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 â b2 = (a + b)(a â 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 â b3 = (a â 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 â 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 â b4 = (a â 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 â b5 = (a â 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 â an-2b + an-3b2 - âĻ - abn-2 + bn-1).
8. If n is even, then an â bn = ( a - b)(an-1 + an-2b + an-3b2 + âĻ + abn-2 + bn-1). an + bn = ( a + b)(an-1 â an-2b + an-3b2 - âĻ + abn-2 - bn-1).
