Algebraic Identities-RRB
For example, the identity holds for all values of x and y. (x+y)2 = x2 + 2xy + y2
Since an identity holds for all values of its variables, it is possible to substitute instances of one side of the equality with the other side of the equality. For example, because of the identity above, we can replace any instance of (x+y)2 with x2 + 2xy +y2 and vice versa.
Clever use of identities offers shortcuts to many problems by making the algebra easier to manipulate. Below are lists of some common algebraic identities.
Binomial Theorem Identities
The following identities are product formulas that are examples of the binomial theorem:
(x+y)2 = x2 + 2 xy + y2
(x-y)2 = x2 – 2 xy + y2
(x+y)3 = x3 + 3 x2 y +3 xy2 + y3
(x-y)3 = x3 – 3 x2 y +3 xy2 – y3
(x+y)4 = x4 + 4 x3 y + 6 x2 y2 +4 xy3 + y4
(x-y)4 = x4 – 4 x3 y + 6 x2 y2 – 4 xy3 + y4
Factoring Identities
The following identities are factoring formulas;
x2 – y2 = (x+y)(x-y)
x3 – y3 = (x-y)(x2 + xy + y2)
x3 + y3 = (x-y)(x2 – xy + y2)
x4 – y4 = (x2 – y2)(x2 + y2)
Three-variable Identities
The following identities can be derived by some clever factoring and manipulation of the terms:
(x +y)(x+ z)(y+z) = (x +y+ z)(xy +xz+ yz)
x3 + y3 + z3 -3 x y z = (x + y + z)(x2 + y2 + z2 – xy -xz -yz)
Example-
The identity 4(x+7)(2 x – 1) = A x² + Bx + C holds for all real values of x. What is A + B + C ?
Ans-
Multiplying the left side of the identity, we have:
4(x+7)(2 x – 1) = 4 (2 x² -x +14 x -7) = 8 x² +52 x – 28
⇒ 8 x² +52 x – 28 = A x² + B x + C
⇒ A = 8, B = 52, C = -28
Hence A + B + C = 8+ 52 -28 = 32