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Algebraic Identities - Definition, Example, Verification, Verification of Algebraic Identities, Binomial Theorem and Identities Table

 

Algebraic identities can be defined as algebraic equations that are always valid for whatever amount of their variables. Algebraic identities are used in the solution of polynomials. On both sides of the equation, variables and constants are present. The left side of an algebraic identity is equal to the right side of the equation. Algebraic identities are therefore utilized in the calculation of algebraic expressions and the solution of various polynomials. You might already have studied a handful of them in rudimentary classes. In this post, we will review them and present you with some more common algebraic identities and provide illustrations.
Difference between an algebraic expression and algebraic identities
An algebraic expression has variables and constants. Variables in expressions can have any value. As a result, if the variable values change, the expression value may change. However, the algebraic identity is symmetric, which makes sense for all continuous variables. There is an equal sign between the terms of an algebraic identity. For an equation to be an algebraic identity, both sides of the equality sign must pertain to a similar value. If the value of the expression is, let’s say, 5 on the left side, then the value of the algebraic identity on the right ought to be 5. The algebraic identity can also be solved using various methods to identify the values of the unknown variables.

Verification of Algebraic Identities

The Substitution Method

  • Substitution typically refers to the substitution of numbers or values for variables or characters.
  • The replacement technique performs a mathematical calculation by replacing the values of variables.
  • Let’s assume we have an equation 3x-9 = 6, and let us substitute the value of x with 4, therefore the equation changes to 3*4 – 9 = 12 – 9 = 3, hence x = 4 is the wrong substitution. Now, on substituting the value of x with 5, we get, 3*5 – 9 = 15 – 9 = 6. Therefore x = 5 is the correct substitution for our equation because the left-hand side of the equation becomes equal to the right-hand side of the equation.

The Activity Method

  • The algebraic identity is geometrically proven using different values of x and y in this technique.
  • The identities are confirmed in the activity approach by clipping and pasting paper.
  • To use this technique of identity verification, you must have a basic understanding of Geometry.

List of Standard Algebraic Identities

All the algebraic identities are derived from a single theorem known as the binomial theorem. Some of the important identities that must be remembered all the time are listed below:

Identities:  
Identity I: (k + l)2 = k2 + 2kl + l2
Identity II: (k - l)2 = k2 - 2kl + l2
Identity III: k2 – l2= (k + l) (k – l)
Identity IV: (x + k) (x + l) = x2 + (k + l) x + kl
Identity V: (k + l + m)2 = k2 + l2 + m2 + 2kl + 2lm + 2mk
Identity VI: (k + l)3 = k3 + l3 + 3kl (k + l)
Identity VII: (k – l)3 = k3 – l3 – 3kl (k – l)
Identity VIII: k3 + l3 + m3 – 3klm = (k + l + m) (k2 + l2 + m2 – kl – lm – mk
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