Algebraic Identities

Algebraic identities are an important set of formulas in maths. They form the foundation working principle of algebra and are helpful to perform computations in simple and easy steps. Certain algebraic problems require working across numerous mathematical step to obtain the answer. Here, with the use of algebraic identities, we are able to perform the calculations without any additional steps. Many of the algebraic identities have been obtained from the binomial expansion of terms. 

An algebra identity means that the left-hand side of the equation is identically equal to the right-hand side, for all values of the variables. Here we shall try to acquaint ourselves with all the algebraic identities, their proofs, and how to use these identities in our math calculations.

Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation. Unlike algebraic expressions, algebraic identities satisfy all the values of the variables. Let us consider an example to understand this better. Consider the equations: 5x - 3 = 12,  10x - 6 = 24, and x² + 5x + 6 = 0. These equations satisfy only a unit value and do not work for any other values of the variables. Now let us consider an equation x²  - 9 = (x + 3)(x - 3). These equations satisfy any values of x. 

From this, we can clearly understand that an algebra identity has an expression that satisfies any values for the variables. The basic algebraic identities are primarily helpful to work out the numerous math problems. The four basic algebraic identities are as follows.

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b²
• (x + a)(x + b) = x² + x(a + b) + ab

IThe binomial expansion based on the binomial theorem is helpful to find derive all the algebraic identities. Let us gain a brief idea of the binomial theorem expansion, which would be helpful to derive algebraic identities of higher degrees. Generally, binomial expansion is used for expansion involving two variables and with n degree. The total number of terms of the binomial expansion is (n + 1) and in the expanded form the power of the variable is reducing, and the power of the second variable is increasing in sequential terms.  Each of the terms in the binomial expansion has a coefficient associated with it.

The following algebraic identities have been derived from the binomial expansion for sum and difference of variable and for a maximum power of 3 the algebraic identities have been listed below as formulas. Further algebraic identities of higher degrees and more variables can also be derived using the above binomial expansion formula.

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b²
• (a + b)³ = a³ +3a²b + 3ab² + b³
• (a - b)³ = a³ - 3a²b + 3ab² - b³
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

Algebraic identities are greatly helpful in easily factorizing algebraic expressions. The given expression representing the expanded form of the formula can be transformed and written as a set of factors using the below algebraic identities. Some of the higher algebraic identities such as a⁴ - b⁴ can be easily derived using the basic algebraic identities a² - b². The below list presents a set of algebraic identities helpful for the factorization of polynomials.

• a² - b² = (a - b)(a + b)
• x² + x(a + b) + ab = (x + a)(x + b)
• a³ - b³ = (a - b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² - ab + b²)
• a⁴ - b⁴ = (a - b)(a + b)(a² + b²)

The algebraic identities for three variables also has been derived using the binomial expansion formula. Further these identities are helpful to easily work across the algebraic expressions with the least number of steps. 

• a² + b² + c² = (a + b + c)² - 2(ab + bc + ac)
• a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - ca - bc)
• (a + b)(a + c)(b + c) = (a + b + c)(ab + ac + bc) - 3abc

Apart from these simple algebraic identities listed above, there are other algebraic identities that we will use in higher grades.

Check them on the page Algebraic Identities Formula and Examples.

The following proofs of algebraic identities will help us to visually understand each of the identities and better understand it. Let us look at the proofs of each of the basic algebraic identities.

Proof of (x + a)(x + b) = x² + x(a + b) + ab

(x+a)(x+b)  is nothing but the area of a rectangle whose sides are (x+a) and (x+b) respectively. The area of a rectangle with sides (x+a) and (x+b) in terms of the individual areas of the rectangles and the square is x², ax, bx , ab.  Summing all these areas we have x² + ax + bx + ab.  This gives us the proof for the algebra identity (x + a)(x + b) = x² + ax + bx + ab = x² + x(a + b) + ab.

Proof of (a + b)² = a² + 2ab + b²

The algebraic expression (a+b)² is nothing but (a+b) × (a+b). This can be visualized as a square whose sides are (a+b) and the area is (a+b)². The square with a side of (a + b) can be visualized as four areas of a², ab, ab, b². The sum of these areas a² + ab + ab + b² gives the area of the square (a+b)². The area of the square (a+b)² =  a² + ab + ab + b² proves the algebraic identity.

Proof of (a + b)(a - b) = a² - b²

The objective is to find the value a² - b², which can be taken as the difference of the area of two squares of sides a units and b units respectively.  This is equal to the sum of are areas of two rectangles as presented in the below figure. One rectangle has a length of a units and a breadth of (a - b) units. Another rectangle is taken with a length of (a - b) and a breadth of b units. Further, we take the areas of the two rectangles and sum the areas to obtain the resultant values. The respective areas of the two rectangles are (a - b) × a = a(a - b) , and (a - b) × b = b(a - b). Finally, we take the sum of these areas to obtain the resultant expression.

a(a + b) + b(a - b) = (a + b)(a - b)

Re-arranging the individual squares and rectangles, we get:

\( (a+b)(a-b)  =a^{2}-b^{2} \)

Proof of (a − b)² = a² −2 ab + b²

Once again, let’s think of (a - b)² as the area of a square with length (a - b). To understand this, let's begin with a large square of area a². We need to reduce the length of all sides by b, and it becomes a - b. We now have to remove the extra bits from a² to be left with (a - b)². In the figure below, (a - b)² is shown by the blue area. To get the blue square from the larger orange square, we have to subtract the vertical and horizontal strips that have the area ab. However, removing ab twice will also remove the overlapping square at the bottom right corner twice. Hence, we add b². Thus we have  (a − b)² = a² − ab − ab + b². Hence this proves the algebraic identity (a − b)² = a² − 2ab + b²

Related Links

• Variable Expressions
• Algebraic Expressions
• Algebraic Formulas
• Linear Equations
• Exponents
• Polynomials
• Quadratic Equations

Who Discovered Algebraic Identities?

The first form of algebraic identities was referred to as the theory of equations. The concepts of algebra were derived by a Persian mathematician and the numerous terms of algebra have been derived from the Arabic language. The birth of algebra can also be attributed to the Babylonians. The various forms of algebra are Rhetorical algebra, syncopated algebra, and symbolic algebra.

How Many Algebraic Identities are there in Maths?

There are four basic algebraic identities in maths. These four identities are helpful in performing numerous calculation and is also useful in deriving numerous other identities. The four identities are as follows.

• (a + b)² = a² + 2ab + b²
• (a + b)² = a² + 2ab + b²
• (a + b)(a - b) = a^2ic - b²
• (x + a)(x + b) = x² + x(a + b) + ab


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What Is the Difference Between Algebraic Identities and Algebra Expressions?

There is a very simple difference between algebraic identities and algebra expressions. In an algebra identity, we have an equals sign and with an expression on either side. And in an algebraic expression, we do not have equals to sign, and the expression results in different values based on different input values of the variables. An example of algebra expression is (a + b)² = a² + 2ab + b², and an example of algebraic expression is f(x) = ax² + bx + c

What are the four algebraic identities?

The four commonly used algebraic identities are given below:

• Square of the sum of two binomials
       \(\begin{align} (a+b)(a+b) = (a + b)^2  =  a^2 + 2ab + b^2 \end{align}\)
• Square of the difference of two binomials
       \(\begin{align}(a-b)(a-b) = (a - b)^2   =   a^2 - 2ab + b^2 \end{align}\)
• Product of the sum and the difference of two binomials
       \(\begin{align} (a + b)(a − b) = a^2 − b^2 \end{align}\)
• Product of two binomials
       \(\begin{align} (𝑥 + a) (x+ b) = x^2 + x(a+b) + ab \end{align}\)

Use the Algebraic Identities Calculator to solve some algebraic identities problems .

How Can we Verify an Algebra Identity?

The algebraic identities can be easily verified in two ways. One is by substituting the values in places of the variables within the algebraic identities. An algebra identity has some expression on either side of the equals to sign.  Here we can substitute the values on either side of the equals to sign and try to obtain the same answer on both sides. Another method is to solve algebraically to verify the algebra identity by manipulating and simplifying the left-hand side of the equation, to obtain the right-hand side of the equation.

What Are the Uses of Algebraic Identities?

The algebraic identities have numerous applications in all the areas of math. The topic of algebra is completely based on these algebraic identities. Also, topics such as geometry, coordinate geometry, trigonometry, calculus have extensive use of algebraic identities.  These algebraic identities are helpful to find to solve problems in simple and easy steps.

How can I Learn Algebraic Identities?

The algebraic identities can be easily learned through the below two simple ways:

• Algebraic identities can be easily memorized by visualizing the identities as a square or rectangle.
• They can also be remembered by the factored forms rather than the simplified forms.

What Is the Formula of Identity?

An identity is an equation that is always true regardless of the value that is assigned to the variable. The simple algebraic identities or formulas are:

• (a + b)² = a² + 2ab + b²
• (a + b)² = a² + 2ab + b²
• (a + b)(a - b) = a² - b²
• (x + a)(x + b) = x² + x(a + b) + ab