Algebraic Identities

Algebraic identities are simple, reliable equations that hold true for all values of their variables. This guide presents the standard algebraic identities, provides clear proofs, and demonstrates their application through worked examples and exercises. Mastery of these identities enables efficient expansion, factorization, and simplification, and serves as an essential foundation for higher-level algebra and problem solving.

Table Of Contents

• What are Algebraic Identities?
• Complete List of Algebraic Identities
• Proofs of the Four Fundamental Identities
• Using Identities for Factorisation
• Solved Examples of Algebraic Identities
• Importance of Algebraic Identities
• Frequently Asked Questions on Algebraic Identities

What are Algebraic Identities?

Algebraic identities are equations that are true for every possible value of the variables not just some values. An algebraic identity is an equation involving variables where the Left Hand Side (LHS) equals the Right Hand Side (RHS) for all values of the variables. It doesn't matter whether you substitute 0, 1, −5, or 1000 the identity always holds true.

Imagine you need to calculate 103². You could multiply 103 × 103 the long way. Or, you could think of it as (100 + 3)², use the identity (a + b)² = a² + 2ab + b², and solve it in about five seconds: 10000 + 600 + 9 = 10609.

Complete List of Algebraic Identities

Standard Algebraic Identities

Two Variable Identities

The following are algebraic identities  are based on two variables and are widely used in mathematics.

• $(a + b)^2 = a^2 + 2ab + b^2$

• $(a - b)^2 = a^2 - 2ab + b^2$

• $(a + b)(a - b) = a^2 - b^2$

• $(a + b)^3 = a^3 + 3a^{2}b + 3ab^2 + b^3$

• $(a - b)^3 = a^3 - 3a^{2}b + 3ab^2 - b^3$

Three Variable Identities 

The following are algebraic identities based on three variables.

• $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$

• $a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ac)$

• $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ca - bc)$

• $(a + b)(b + c)(c + a) = (a + b + c)(ab + ac + bc) - 2abc$
  
  

Proofs of the Four Fundamental Identities

1. Identity I: (a + b)² = a² + 2ab + b²

Algebraic Proof of (a + b)² = a² + 2ab + b²

(a + b)² = (a + b) × (a + b) [rewrite square as product]

= a(a + b) + b(a + b) [distribute the first bracket]

= a² + ab + ba + b² [distribute again]

= a² + ab + ab + b² [since ab = ba by commutativity]

= a² + 2ab + b²

Visual Proof: 

Consider a square with side (a+b). Its area i $s (a+b)^2.$ If we divide this square into smaller regions, the total area becomes  $a^2+ab+ab+b^2$, which simplifies to $a^2+2ab+b^2$.

Since the total area is unchanged, we get the identity:

$(a+b)^2 = a^2+2ab+b^2$

2. Identity II: (a − b)² = a² − 2ab + b²

Algebraic Proof of (a − b)² = a² − 2ab + b²

(a − b)² = (a − b)(a − b) [rewrite]

= a(a − b) − b(a − b) [distribute]

= a² − ab − ba + b² [expand]

= a² − ab − ab + b² [ab = ba]

= a² − 2ab + b²

Visual Proof: 

Consider a square of side a. Its area is a². 2. Now, remove a smaller square of side b from it, along with the two rectangular regions of dimensions a×b.

The remaining area can be written as a²−ab−ab+b², which simplifies to  $a^2−2ab+b^2$.

Since the total area is preserved through this decomposition, we obtain:

$(a−b)^2=a^2−2ab+b^2.$

3. Identity III: (a + b)(a − b) = a² − b²

Algebraic Proof of (a + b)(a − b) = a² − b²

(a + b)(a − b) = a(a − b) + b(a − b) [distribute]

= a² − ab + ba − b² [expand]

= a² − ab + ab − b² [ab = ba]

= a² − b² [the −ab and +ab cancel out]

Visual Proof: 

To prove the identity, we consider a square and remove a smaller square of area b^2 from it. The remaining region can be divided into two rectangles.

The total remaining area is the sum of these two rectangles:

a(a−b)+b(a−b)

Taking (a−b) common, we get:

(a−b)(a+b)

On the other hand, the remaining area is also equal to $a^2−b^2.$

Since both expressions represent the same area, we conclude:

$(a−b)(a+b)=a^2−b^2$

4. Identity IV: (x + a)(x + b) = x² + (a + b)x + ab

Algebraic Proof:

(x + a)(x + b) = x(x + b) + a(x + b) [distribute]

= x² + bx + ax + ab [expand]

= x² + (a + b)x + ab  [combine bx + ax]

Visual proof: To prove this visually, consider a rectangle of sides (x+a) and (x+b).

Split it into four parts:  $x^2$, ax, bx, and ab.

Adding these areas gives  $x^2+(a+b)x+ab$, hence  $(x+a)(x+b)=x^2+(a+b)x+ab$.

Using Identities for Factorisation

The identities can also be used them to factorise (break down) complex-looking expressions into simpler factors.

Give below are a few examples of using identities for factorisation.

Solved Examples of Algebraic Identities

Example 1: Expand (3x + 4y)²

Solution: using Identity I: (a+b)² = a² + 2ab + b²

Identify a = 3x, b = 4y

(3x + 4y)² = (3x)² + 2(3x)(4y) + (4y)²

= 9x² + 24xy + 16y²

Example 2: Find the product 998 × 1002 using identities

Solution: using Identity III: (a+b)(a−b) = a² − b²

Write 998 = 1000 − 2 and 1002 = 1000 + 2

So 998 × 1002 = (1000 − 2)(1000 + 2)

= 1000² − 2² = 1000000 − 4 = 999996

Example 3: If x + y = 10 and xy = 21, find x² + y²

Solution: We know (x + y)² = x² + 2xy + y²

So x² + y² = (x + y)² − 2xy

= (10)² − 2(21) = 100 − 42 = 58

Example 4: If a + b + c = 0, find the value of a³ + b³ + c³

Solution: Using: a³ + b³ + c³ − 3abc = (a+b+c)(a²+b²+c²−ab−bc−ca)

Since a + b + c = 0, the RHS becomes 0 × (anything) = 0

Therefore a³ + b³ + c³ − 3abc = 0

∴ a³ + b³ + c³ = 3abc

Example 5: Factorise 25x² − 70xy + 49y²

Solution: 25x² = (5x)², 49y² = (7y)², and 70xy = 2 × 5x × 7y

This matches (a − b)² = a² − 2ab + b², where a = 5x and b = 7y

∴ 25x² − 70xy + 49y² = (5x − 7y)²

Importance of Algebraic Identities

Learning algebraic identities is very important because they:

• Help to simplify algebraic expressions

• Make calculations faster and easier

• Support factoring and expansion

• Help solve complex equations

• They are used in geometry, physics, and higher-level math

Understanding algebraic identities also builds a strong foundation for more advanced topics in algebra, such as quadratic equations, polynomials, and calculus.

 

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