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Standard Identities in Maths
There are a number of standard identities in mathematics that are used to simplify calculations. Some of the most common identities are the Pythagorean theorem, the commutative and associative laws of addition, and the distributive law.
What are Standard Identities in Maths?
Standard identities are mathematical relationships between certain mathematical quantities which are always true. They include the Pythagorean theorem, the basic trigonometric functions, and the fundamental theorem of calculus.
Algebraic Identities
An algebraic identity is a statement in mathematics that is always true. There are a number of different algebraic identities that are used in various mathematical proofs. Some of the most common algebraic identities are the commutative, associative, and distributive laws.
Standard Algebraic Identities Under Binomial Theorem
The binomial theorem states that for any real number “x” and any positive integer “n”,
The Binomial Theorem is a statement of the following form:
For any real number “x” and any positive integer “n”,
The left side of the equation is a polynomial in “x” of degree “n”. The right side of the equation is the product of “n” factors, each of which is a binomial in “x” and “y”.
Standard Algebraic Identities Under Factoring
The following are the standard algebraic identities under factoring:
1) The product of two binomials is the sum of their products of the individual terms.
For example, the product of (x + y) and (x – y) is x² – y².
2) The difference of two binomials is the sum of their differences of the individual terms.
For example, the difference of (x + y) and (x – y) is 2x.
3) The sum of two binomials is the sum of their individual terms.
For example, the sum of (x + y) and (x – y) is 2x.
4) The product of a binomial and a trinomial is the sum of the products of the individual terms.
For example, the product of (x + y) and (x² + y²) is x³ + 2x²y² + y³.
Standard Algebraic Identities in Three – Variables
The following list of algebraic identities in three variables are some of the most common ones.
1. a + b + c = 0
2. a.b = b.a
3. a.(b + c) = ab + ac
4. (a + b) . (c + d) = ac + ad + bc + bd
What are Standard Identities in Maths?
Standard identities are identities that are introduced by multiplying one binomial with another binomial.
Example,
let us consider (p+q)(p +q) or (p +q)²
(p + q)² = (p + q)(p + q)
(p + q)(p + q) or (p + q)²
p² + pq + pq + q²
p² + q² + 2pq (as pq = qp)
Hence, (p + q)² = p² + q² + 2pq (1)
Clearly, we can see this is an identity as the expression on LHS is obtained from the RHS. One may verify that for any values of p and q, the values of two sides will always be equal.
Next, we consider, (p – q)(p – q) or (p – q)²
We can say, (p – q)² = (p – q)(p – q)
p(p – q) q(p – q)
p² – pq – pq + q²
p² + q² – 2pq as (pq = qp)
Hence, (p – q)² = p² + q² – 2pq (2)
Next, we consider (p + q)(p – q). We can say, (p + q)(p – q)= p(p – q) + q(p – q)
(p² – pq + pq – q²) = p² – q² (as pq = qp)
(p + q)(p – q) = p² – q² (3)
Finally, we consider (p + q) (p + r) We can say, (p + q) (p + r) = p² + pr + pq + qr
or
(p + q) (p + r) = p² + (q + r)p +pq (4)
The above-given identities 1, 2, 3, 4 are considered as standard identities.
Algebraic Identities
An is an equation that is valid for any value of its variables.
For example, the identity (p + q)² = p² + 2pq + q² is valid for all values of p and q.
As an identity holds valid for all values of its variables, it is possible to replace one side of equality with another side of equality. For example, in the above-given identity, we can easily substitute instances of (p + q)² = p² + q² + 2pq and vice versa.
Using identity in an intelligent way offers shortcuts to many problems by making algebra easier to operate. Below is a list of some standard algebraic identities.
Standard Algebraic Identities Under Binomial Theorem
Following are some of the standard identities in Algebra under binomial theorem.
Identity 1: (p + q)² = p² + 2pq + q²
Identity 2: (p – q)² = p² + q² – 2pq
Identity 3: (p + q)³ = p³ + q³ + 3pq(p + q)
Identity 4: (p – q)³ = p³ – q³ – 3pq(p – q)
Identity 5: (p + q)⁴ = p⁴ + 4p³q + 6p²q² + 4pq³ + q⁴
Identity 6: (p – q)⁴ = p⁴ – 4p³q + 6p²q² – 4pq³ + q⁴
Standard Algebraic Identities Under Factoring
Following standard algebraic identities are factoring formula:
Identity 1: p² – q² = ( p + q)(p – q)
Identity 2: p³ + q³ = (p + q)(p² + q² – pq)
Identity 3: p³ – q³ = ( p – q)(p² + q² + pq)
Identity 4: p⁴ – q⁴ = (p² + q²)(p² – q²)
Standard Algebraic Identities in Three – Variables
Following standard identities in three- variables are obtained by some factoring and manipulation of terms.
Identity 1: (p + q)(p + r)(q + r) = (p + q + r)(pq + pr +qr) – pqr
Identity 2: p² + q² + r² = (p + q + r)² – 2 (pq + pr + qr)
Identity 3: p + q + r – 3pqr = (p + q + r)(p² + q² + r² – pq – pr – qr)
Solved Examples
1. Using identity, (p + q)² = p² + q² + 2pq
Find (2m + 3n)²
Solution:
As we know, (p + q)² = p² + q² + 2pq
Accordingly, (2m + 3n)² = (2m)² + (3n)² + 2(2m)(3n)
Hence, (2m + 3n)² = 4m² + 9n² + 12mn
2. Using identity, (p – q²) = (p² + q² – 2pq)
Find (2m – 3n)²
Solution:
As we know, (p – q²) = (p² + q² – 2pq)
Accordingly, (2m – 3n)² = (2m)² +(3n)² – 2(2m)(3n)
Hence, (2m + 3n)² = 4m² + 9n² – 12mn
3. Using Identity,(p + q)(p – q) = p² – q²
Find (2m + 3n) (2m – 3n)
Solution:
As we know, (p + q)(p – q) = p² – q²
Accordingly, (2m + 3n)(2m – 3n) = (2m)² – (3n)²
(2m + 3n)(2m – 3n) = 4m² – 9n²
