An algebraic identity is a specialkind of equation in which the LHS and RHS are equal for every value ofthe variable. The difference between an equation and an identity will become clear with the help of the iven table below.
| Equation | Identity |
|---|---|
| Consider an equation 2x + 5 = 13. In this equation, LHS and RHS are equal only for x = 4 (which is the solution for this equation). If x not equal to 4, 2x + 5 is not equal to 13. |
Consider an identity (a + b)(a - b) a2 - b2 Here LHS and RHS are equal for every value of a and b. For example, take a = 4 and b = 1. Then, LHS = (a + b)(a - b) = (4 + 1)(4 - 1) = 5x3 = 15 and RHS = a2 - b2 = 42 - 12 = 16 - 1 = 15. Thus, LHS = RHS + 15. |
| An equation is true only for some values of the variables in it. |
An identity is true for every value of the variables in it. |
There are four identites which are very useful in solving the problems of algebra (based on constants and variables).
These identities are as follows:
Verification: (a + b)2 = (a + b)(a + b)
= a(a + b ) + b(a +b)
= a x a + a x b +b x a + b x b
= a2 + ab +b a + b2
= a2 + ab +ab + b2
= a2 + 2ab + b2
Verification: (a - b)2 = (a - b)(a - b)
= a(a - b ) - b(a -b)
= a x a - a x b +b x a - b x b
= a2 - ab +b a - b2
= a2 - ab -ab + b2
= a2 - b2
Verification: (a + b)(a - b) = a(a - b) + b(a - b)
= a x a - a x b +b x a - b x b
= a2 - ab +ba - b2
= a2 - ab +ba - b2
= a2 - b2
Verification: (x + a)(x + b) = x(x + b) + a(x + b)
= x x x + x - b + a x x + a x b
= x2 + xb + a x x + a x b
= x2 + bx + a x x + a x b
= x2 + (a + b)x + a x b
Let us move on to another topic, just click here! Lesson 7 Division of Polynomials
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