What are Algebraic Identities(With Definition, Types & Derivations)

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In mathematics, an identity is an equality that remains true even if you change all the variables that are used in that equality. Algebraic identities are used as formulas in math that help to perform computations in simple and easy steps. For example, $(a + b)^{2} = a^{2} + 2ab + b^{2}$ is an algebraic identity.

Let’s understand what is an identity in algebra and what are its different types.

What are Algebraic Identities?

Algebraic identities are equations where the value of the left-hand side of the equation is always equal to the value of the right-hand side. They are satisfied with any values of the variables. 

Let’s consider an example to understand this better. 

Consider an equation $2x – 7 = 3$. This equation is true only for one value of $x$ and it is $x = 5$.

For $x = 3$, LHS = $2 \times 5 – 7 = 10 – 7 = 3 = $ RHS.

If we take any other value say $x = 2$, then for $x = 2$, LHS = $2 \times 2 – 7 = 4 – 7 = -3 \ne $ RHS.

Consider another equation $x^{2} + 5x – 14 = 0$. This equation is true only for two values of $x$ and they are $x = 2$ and $x = -7$.

For $x = 2$, LHS = $2^{2} + 5 \times 2 – 14 = 4 + 10 – 14 = 0 = $ RHS, and similarly, for $x = -7$, LHS = $\left(-7 \right)^{2} + 5 \times \left(-7 \right) – 14 = 49 – 35 – 14 = 0$.

For any other value, the LHS and RHS of $x^{2} + 5x – 14 = 0$ are not equal.

Now, consider an identity $\left(a + b \right)^{2} = a^{2} + 2ab + b^{2}$. It is called an identity, because the LHS and RHS of this equation are always the same irrespective of the values of the variables $a$ and $b$.

For $a = 2$ and $b = 3$, LHS = $\left(2 + 3 \right)^{2} = 5^{2} = 25$ and RHS = $2^{2} + 2 \times 2 \times 3 + 3^{2}$

$ = 4 + 12 + 9 = 25$

Similarly, for another set of values for $a = 4$ and $b = 6$, LHS = $\left(4 + 6 \right)^{2} = 100$ and RHS =  $4^{2} + 2 \times 4 \times 6 + 6^{2} = 16 + 48 + 36 = 100$.

Types of Algebraic Identities

There are various types of algebraic identities depending on the number of variables present in an identity. In this section, we’ll discuss two broad categories of identities.

• Two-Variable Identities
• Three-Variable Identities

Two-Variable Identities

These are the identities that contain two variables. These identities can be easily verified by expanding the square/cube and doing polynomial multiplication.  The most common two-variable identities are

• $\left(a + b \right)^{2} = a^{2} + 2ab + b^{2}$
• $\left(a – b \right)^{2} = a^{2} – 2ab + b^{2}$
• $\left(a + b \right) \left(a – b \right) = a^{2} – b^{2}$
• $\left(a + b \right)^{3} = a^{3} +3a^{2}b + 3ab^{2} + b^{3}$
• $\left(a – b \right)^{3} = a^{3} – 3a^{2}b + 3ab^{2} – b^{3}$
• $a^{2} – b^{2} = \left(a – b \right) \left(a + b \right)$
• $x^{2} + x \left(a + b \right) + ab = \left(x + a \right) \left(x + b \right)$
• $a^{3} – b^{3} = \left(a – b \right) \left(a^{2} + ab + b^{2} \right)$
• $a^{3} + b^{3} = \left(a + b \right) \left(a^{2} – ab + b^{2} \right)$

Three-Variable Identities

These are the identities that contain three variables. These identities are helpful to easily work across algebraic expressions with the least number of steps.  The most common three-variable identities are

• $\left(a + b + c \right)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac$
• $a^{2} + b^{2} + c^{2} = \left(a + b + c \right)^{2} – 2\left(ab + bc + ac \right)$
• $a^{3} + b^{3} + c^{3} – 3abc = \left(a + b + c \right) \left(a^{2} + b^{2} + c^{2} – ab – ca – bc \right)$

Proofs of Identities

We can prove these identities using simple algebraic methods. Following are the proofs of the two and three-variable identities.

Proofs of Two-Variable Identities

1. $\left(a + b \right)^{2} = a^{2} + 2ab + b^{2}$

LHS of the identity is $\left(a + b \right)^{2}$

$\left(a + b \right)^{2} = \left(a + b \right)\left(a + b \right) = a\left(a + b \right) + b\left(a + b \right) = a \times a + a \times b + b \times a + b \times b $

$= a^{2} + ab + ba + b^{2} = a^{2} + ab + ab + b^{2} = a^{2} + 2ab + b^{2}$

Let’s understand the formula geometrically.

Consider a square of edge length $a$ units (Shown orange in the figure)

Let’s further increase the edge length of the square by $b$ units, so that the edge length of the new square(big) becomes $\left(a + b \right)$.

Area of new square of edge length = Area of orange square + Area of two cyan squares + Area of the green square

= $a^{2} + 2ab + b^{2}$.

Since the edge length of the big square is $\left(a + b \right)$, therefore, area of the big square is $\left(a + b \right)^{2}$.

Thus, we get $\left(a + b \right)^{2} = a^{2} + 2ab + b^{2}$

Examples

Ex 1: Find the value of $\left(2x + 5y \right)^{2}$ using $\left(a + b \right)^{2}$ formula.

Comparing $\left(2x + 5y \right)^{2}$ with $\left(a + b \right)^{2}$, we get $a = 2x$ and $b = 5y$

$\left(a + b \right)^{2} = a^{2} + 2ab + b^{2}$

Therefore, $\left(2x + 5y \right)^{2} = \left(2x \right)^{2} + 2\times 2x \times 5y + \left(5y \right)^{2} = 4x^{2} + 20xy + 25y^{2}$.

Ex 2: Evaluate $107^{2}$ using $\left(a + b \right)^{2}$ formula.

$107^{2}$ can be written as $\left(100 + 7 \right)^{2}$.

Comparing $\left(100 + 7 \right)^{2}$ with $\left(a + b \right)^{2}$, we get $a = 100$ and $b = 7$

Therefore, $\left(100 + 7 \right)^{2} = 100^{2} + 2 \times 100 \times 7 + 7^{2} = 10000 + 1400 + 49 = 11449$

Thus,  $107^{2} = 11449$.

2. $\left(a – b \right)^{2} = a^{2} – 2ab + b^{2}$

LHS of the identity is $\left(a – b \right)^{2}$

$\left(a – b \right)^{2} = \left(a – b \right)\left(a – b \right) = a\left(a – b \right) – b\left(a – b \right) = a \times a + a \times \left(-b \right) – b \times a – b \times \left(-b \right) $

$= a^{2} – ab – ba + b^{2} = a^{2} – ab – ab + b^{2} = a^{2} – 2ab + b^{2}$

3. $\left(a + b \right) \left(a – b \right) = a^{2} – b^{2}$

LHS of the identity is $\left(a + b \right) \left(a – b \right)$

$\left(a + b \right) \left(a – b \right) = a\left(a – b \right) + b\left(a – b \right) = a \times a + a \times \left(-b \right) + b \times a – b \times b$

$ = a^{2} – ab + ba – b^{2} = a^{2} – ab + ab – b^{2} = a^{2} – b^{2}$

4. $\left(a + b \right)^{3} = a^{3} +3a^{2}b + 3ab^{2} + b^{3}$

LHS of the identity is $\left(a + b \right)^{3}$

$\left(a + b \right)^{3} = \left(a + b \right)\left(a + b \right)^{2} = \left(a + b \right) \left(a^{2} + 2ab + b^{2} \right) = a\left(a^{2} + 2ab + b^{2} \right) + b\left(a^{2} + 2ab + b^{2} \right)$

$= a \times a^{2} + a \times 2ab + a \times b^{2} + b \times a^{2} + b \times 2ab + b \times b^{2} = a^{3} + 2a^{2}b + ab^{2} + a^{2}b + 2ab^{2} + b^{3}$

$ = a^{3} + 2a^{2}b + a^{2}b + ab^{2} + 2ab^{2} + b^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$

5. $\left(a – b \right)^{3} = a^{3} – 3a^{2}b + 3ab^{2} – b^{3}$

LHS of the identity is $\left(a – b \right)^{3}$

$\left(a – b \right)^{3} = \left(a – b \right)\left(a – b \right)^{2} = \left(a – b \right) \left(a^{2} – 2ab + b^{2} \right) = a\left(a^{2} – 2ab + b^{2} \right) – b\left(a^{2} – 2ab + b^{2} \right) $

$= a^{3} – 2a^{2}b + ab^{2} – a^{2}b + 2ab^{2} – b^{3} = a^{3} – 2a^{2}b – a^{2}b + ab^{2} + 2ab^{2} – b^{3}$

$ = a^{3} – 3a^{2}b + 3ab^{2}$ – b^{3}$

6. $x^{2} + x \left(a + b \right) + ab = \left(x + a \right) \left(x + b \right)$

RHS of the identity is $\left(x + a \right) \left(x + b \right)$

$\left(x + a \right) \left(x + b \right) = x\left(x + b \right) + a\left(x + b \right) = x^{2} + bx + ax + ab = x^{2} + ax + bx + ab = x^{2} + \left(a + b \right)x + ab$

7. $a^{3} – b^{3} = \left(a – b \right) \left(a^{2} + ab + b^{2} \right)$

RHS of the identity is $\left(a – b \right) \left(a^{2} + ab + b^{2} \right)$

$\left(a – b \right) \left(a^{2} + ab + b^{2} \right) = a\left(a^{2} + ab + b^{2} \right) – b\left(a^{2} + ab + b^{2} \right) = a^{3} + a^{2}b + ab^{2} – a^{2}b – ab^{2} – b^{3} = a^{3} + a^{2}b – a^{2}b + ab^{2} – ab^{2} – b^{3} = a^{3} – b^{3}$

8. $a^{3} + b^{3} = \left(a + b \right) \left(a^{2} – ab + b^{2} \right)$

RHS of the identity is $\left(a + b \right) \left(a^{2} – ab + b^{2} \right)$

$\left(a + b \right) \left(a^{2} – ab + b^{2} \right) = a\left(a^{2} – ab + b^{2} \right) + b\left(a^{2} – ab + b^{2} \right) = a^{3} – a^{2}b + ab^{2} + a^{2}b – ab^{2} + b^{3} = a^{3} – a^{2}b + a^{2}b + ab^{2} – ab^{2} + b^{3} = a^{3} + b^{3}$

Proofs of Three-Variable Identities

1. $\left(a + b + c \right)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac$

LHS of the identity is $\left(a + b + c \right)^{2} = \left(a + \left(b + c \right) \right)^{2} = a^{2} + 2a\left(b + c \right) + \left(b + c \right)^{2} = a^{2} + 2ab + 2ac + b^{2} + c^{2} + 2bc = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac$

2. $a^{2} + b^{2} + c^{2} = \left(a + b + c \right)^{2} – 2\left(ab + bc + ac \right)$

RHS of the identity is $\left(a + b + c \right)^{2} – 2\left(ab + bc + ac \right) = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac – 2ab – 2bc – 2ac$

$  = a^{2} + b^{2} + c^{2} + 2ab – 2ab + 2bc – 2bc + 2ac – 2ac = a^{2} + b^{2} + c^{2}$

3. $a^{3} + b^{3} + c^{3} – 3abc = \left(a + b + c \right) \left(a^{2} + b^{2} + c^{2} – ab – ca – bc \right)$

RHS of the identity is $\left(a + b + c \right) \left(a^{2} + b^{2} + c^{2} – ab – ca – bc \right) = a\left(a^{2} + b^{2} + c^{2} – ab – ca – bc \right) + b\left(a^{2} + b^{2} + c^{2} – ab – ca – bc \right) + c\left(a^{2} + b^{2} + c^{2} – ab – ca – bc \right)$

$ = a^{3} + ab^{2} + ac^{2} – a^{2}b – a^{2}c – abc + a^{2}b + b^{3} + bc^{2} – ab^{2} – abc – b^{2}c + a^{2}c + b^{2}c + c^{3} – abc – c^{2}a – bc^{2}$

$ = a^{3} + ab^{2} – ab^{2} + ac^{2} – c^{2}a – a^{2}b + a^{2}b – a^{2}c + a^{2}c + b^{3} + bc^{2} – bc^{2} – b^{2}c + b^{2}c + c^{3} – abc – abc – abc = a^{3} + b^{3} + c^{3} – 3abc$

Difference Between an Algebraic Identity and an Algebraic Equation

Following are the differences between an algebraic identity and an algebraic equation.

Practice Problems

1. When $a + b + c = 0$, what is the value of $a^{3} + b^{3} + c^{3}$?
2. Find the product of $\left(x – 1 \right) \left(x – 1 \right)$ and its value for $x = 3$
3. Find the product of $\left(x + 1 \right) \left(x + 1 \right)$ and its value for $x = 5$
4. Evaluate the following using identities
   • $105^{2}$
   • $97^{2}$
   • $1002^{3}$
   • $998^{3}$

FAQs

Conclusion

Algebraic identities are equations where the value of the left-hand side of the equation is always equal to the value of the right-hand side. They are satisfied with any values of the variables. Algebraic identities are used as formulas in math that help to perform computations in simple and easy steps.

Recommended Reading

• What is the Meaning of Equation – Definition, Types & Examples
• Division of Algebraic Expressions(With Methods & Examples)
• Multiplication of Algebraic Expressions(With Methods & Examples)
• Subtraction of Algebraic Expressions(With Methods & Examples)
• Addition of Algebraic Expressions(With Methods & Examples)
• What is Algebraic Expression(Definition, Formulas & Examples)
• What is Algebra – Definition, Basics & Examples
• What is Pattern in Math (Definition, Types & Examples)
• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)