Algebraic Identities
Algebraic identities are an important set of formulas in math. They form the foundation of algebra and are helpful to perform computations in simple and easy steps. Certain algebra problems require working across numerous mathematical steps to obtain the answer. Here, with the use of algebraic identities, we are able to perform the calculations without any additional steps.
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.
| 1. | What Are Algebraic Identities? |
| 2. | Two Variable Identities |
| 3. | Three Variable Identities |
| 4. | Factorization Identities |
| 5. | Proof of Algebraic Identities |
| 6. | FAQs on Algebraic Identities |
What Are Algebraic Identities?
Algebraic identities are equations in algebra 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. They are satisfied with any 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 for certian value(s) of x and do not work for any value in general. Now let us consider an equation x2 - 9 = (x + 3)(x - 3). Note that this equation is satisfied for any value of x (try to substitute any number for x on both left and right sides, you should be getting the same answer).
These are helpful to work out numerous math problems. The four basic algebra 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
Two Variable Identities
The following are the identities in algebra with two variables. These identities can be easily verified by expanding the square/cube and doing polynomial multiplication. For example, to verify the first identity below, (a + b)2 = (a + b) (a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2. In the same way, we can verify the other identities as well.
- (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
Example: Expand (2x + y)2.
Solution:
To expand the given expression, substitute a = 2x and b = y in (a + b)2 = a2 + 2ab + b2,
(2x + y)2 = (2x)2 + 2(2x)(y) + y2
= 4x2 + 4xy + y2
Three Variable Identities
The algebra identities for three variables also have been derived just the way the two variable identities were. Further, these identities are helpful to easily work across the algebraic expressions with the least number of steps.
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
- 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)(b + c)(c + a) = (a + b + c)(ab + ac + bc) - 2abc
Example: When a + b + c = 0, what is the value of a3 + b3 + c3?
Solution:
By one of the above identities,
a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - ca - bc)
Substituting (a + b + c) = 0, we get
a3 + b3 + c3 - 3abc = 0 (a2 + b2 + c2 - ab - ca - bc)
a3 + b3 + c3 - 3abc = 0
a3 + b3 + c3 = 3abc
Apart from the identities that are listed above, there are other algebraic identities that we will use in higher grades. Check them on the page Algebraic Identities Formula and Examples.
Factorization Identities
Algebraic identities are greatly helpful in easily factorizing algebraic expressions. Using these identities, some of the higher algebraic expressions such as a4 - b4 can be easily factored using the basic algebraic identities like a2 - b2 = (a - b)(a + b). 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)
Example: a4 - b4 = (a2)2 - (b2)2
= (a2 - b2) (a2 + b2)
= (a - b)(a + b)(a2 + b2)
Proof of Algebra Identities
The following proofs of algebra 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, and 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, and b2. The sum of these areas a2 + ab + ab + b2 gives the area of the big square (a+b)2. Hence, (a+b)2 = a2 + ab + ab + b2.
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 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)
Hence, a2 - b2 = (a - b) (a + b)
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
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FAQs on Algebraic Identities
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) = a2ic - b2
- (x + a)(x + b) = x2 + x(a + b) + ab
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.
What are the Four Algebraic Identities?
The four commonly used algebraic identities are given below:
- Square of the sum of two binomials
(a + b)2 = (a+b)(a+b) = a2 + 2ab + b2 - Square of the difference of two binomials
(a - b)2 =( a-b)(a-b) = a2 - 2ab + b2 - Product of the sum and the difference of two binomials
(a + b)(a − b) = a2 − b2 - Product of two binomials
(𝑥 + a) (x+ b) = x2 + x(a+b) + ab
Use the Algebraic Identities Calculator to solve some algebraic identities problems .
What Is the Difference Between Algebra Identities and Algebra Expressions?
There is a very simple difference between algebra identities and algebra expressions. In an algebra identity, we have an equals sign 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 ax2 + bx + c.
How Can we Verify an Algebra Identity?
The algebra 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 areas of math. The topic of algebra is completely based on these algebra identities. Also, topics such as geometry, coordinate geometry, trigonometry, calculus have extensive use of algebraic identities. These are helpful to find to solve problems in simple and easy steps.
Also read
How can I Learn Algebraic Identities?
The algebraic identities can be easily learned through the below two simple ways:
- They can be easily memorized by visualizing the identities as squares or rectangles.
- 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)(a - b) = a2 - b2
- (a + b)2 = a2 + 2ab + b2
- (a + b)2 = a2 + 2ab + b2
- (x + a)(x + b) = x2 + x(a + b) + ab