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Algebra is essentially the practice of performing arithmetic-like operations with non-numerical mathematical objects. However, until the nineteenth century, algebra was primarily concerned with the theory of equations. The fundamental theorem of algebra, for example, belongs to the theory of equations and is no longer considered to be part of algebra.
Definition: –
The algebraic equations that are valid for all values of variables in them are said to be algebraic identities. Polynomial factorization is also done with them. The Algebraic identities can be used in this way to compute algebraic expressions and evaluate various polynomials.
A deeper understanding of algebraic identities helps to improve the efficiency with which problem sums are solved. The factorization of polynomials is one of the most important applications of algebraic identities. We will explore algebraic identities and their properties in this post, along with several examples that will help you better comprehend the notion of algebraic identities. We’ll go through the fundamentals and concentrate on grasping the conceptual components of algebraic identities.
Difference between Algebraic Identities and Equations: –
An algebraic identity is an equation that maintains equality between both sides of the equation regardless of the value of the variables evaluated.
Take a look at the following equation.
6t – 4 = 2
Only if the value of t is 4 this equation is valid. As a result, this is merely an equation rather than an algebraic identity.
Now again have a look at the sample below:
(p + z)² = p² + 2pz + z²
If we take the variables’ values as,
p = 4; z = 2;
Then RHS and LHS will have the same value (i.e. 36).
If p and z are equal, i.e., p = z = 4
Then both RHS and LHS will have the same value (i.e. 64).
As a result, (p + z)² = p² +2pz +z² is true for any value and hence it can be called an algebraic identity.
Standard Algebraic Identities:-
The Binomial Theorem is the source of all standard algebraic identities.
Some of the standard algebraic identities are as follows:
- (M + N)² = M² + 2MN + N²
- (M – N)² = M² – 2MN + N²
- M² – N²= (M + N)(M – N)
- (M + a)(M + b) = M² + (a + b) M + ab
- (M + N + r)² = M² + N² + r² + 2MN + 2Nr + 2Mr
- (M + N)³ = M³ + N³ + 3MN (M + N)
- (M – N)³ = M³ – N³ – 3MN (M – N)Here, p, q and r are the variables and a, b are constants.
Proofs of these identities:-
1. (M + N)² = M² + 2MN + N²
LHS = (M + N)²
= (M + N).(M + N)
= (M*M + M*N + N*M + N*N)
= (M² + MN + MN + N²)
= (M² + 2MN + N²)= RHS
- (M – N)² = M² – 2MN + N²
LHS = (M – N)²
= (M – N).(M – N)
= (M.M – M.N – N.M + N.N)
= (M² – MN – MN + N²)
= (M² – 2MN + N²)= RHS
- M² – N²= (M + N)(M – N)
RHS = (M + N)(M – N)
= (M.M – M.N + N.M – N.N)
= (M² – MN + MN – N²)
= (M² – N²) = LHS
- (M + a)(M + b) = M² + (a + b) M + ab
LHS = (M + a)(M + b)
= (M.M + M.b + a.M + a.b)
= M² + (a + b)M + ab = RHS
- (M + N + r)² = M² + N² + r² + 2MN + 2Nr + 2Mr
LHS = (M + N + r)²
= (M+N)² + r² + 2.(M + N)r
= M² + N² + 2MN + r² + 2Mr + 2Nr
= M² + N² + r² + 2MN + 2Nr + 2Mr = RHS
- (M + N)³ = M³ + N³ + 3MN (M + N)
LHS = (M + N)³
= (M + N)(M + N)(M + N)
= (M² + 2MN + N²)(M + N)
= (M³ + M²N + 2M²N + 2MN² + N²M + N³)
= M³ + N³ + 3M²N + 3MN²
= M³ + N³ + 3MN (M + N) = RHS
- (M – N)³ = M³ – N³ – 3MN (M – N)
LHS = (M – N)³
= (M – N)(M – N)(M – N)
= (M² – 2MN + N²)(M – N)
= (M³ – M²N – 2M²N + 2MN² + N²M- N³)
= M³ – N³ – 3M²N + 3MN²
= M³ – N³ – 3MN (M – N) = RHS
Conclusion:-
A purely mathematical statement is an algebraic expression or an algebraic equation. An equation in algebra is made up of constants, variables, and exponents. When the values of the variables change in an equation, the equality becomes invalid, and the equation can no longer be considered an identity.
An algebraic identity exists when the values of both the right-hand and left-hand sides are always the same, even when the variables’ values are changed. Terms are the building blocks of algebraic expressions. Binomial and trinomial expressions have two and three terms, respectively.
