Standard Algebraic Identities List, Formula and Examples
Algebra is one of the most important chapters of basic mathematics. Students get to know about Algebraic Identities in the lower grades, at the high school level, and then move up to the upper grades and learn higher levels of algebraic Identities. Algebraic identification is a broad topic and is useful in all areas of a student's life. An algebraic identifier is an algebraic equation that applies to all variable values in it. An algebraic equation is a mathematical expression consisting of numbers, variables (unknown values), and mathematical functions (addition, subtraction, multiplication, division, etc.) they are mainly used to find elements of polynomials.
What are Algebraic Identities?
Algebraic identities are equations in algebra where the left-hand side is always equal to the right-hand side, regardless of the values of the variables involved. These identities hold true for all values of the variables. To understand this, consider the equations: 5x - 3 = 12, 10x - 6 = 24, and x^2 + 5x + 6 = 0. These equations are valid only for specific values of x, not universally. However, an equation like x^2 - 9 = (x + 3)(x - 3) is valid for any value of x. Substituting any number for x will give the same result on both sides of the equation.
All Algebraic Identities are extremely useful for solving various mathematical problems. The four Basic Algebraic Identities are:
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
(a + b)(a - b) = a^2 - b^2
(x + a)(x + b) = x^2 + x(a + b) + ab
Standard Algebraic Identities Formula List
Basic Identities
Square of a Sum:
(a + b)² = a² + 2ab + b²
Square of a Difference:
(a - b)² = a² - 2ab + b²
Difference of Squares:
(a + b)(a - b) = a² - b²
Expansion of (x + a)(x + b):
(x + a)(x + b) = x² + x(a + b) + ab
Cubic Identities
Cube of a Sum:
(a + b)³ = a³ + 3a²b + 3ab² + b³
Cube of a Difference:
(a - b)³ = a³ - 3a²b + 3ab² - b³
Sum of Cubes:
a³ + b³ = (a + b)(a² - ab + b²)
Difference of Cubes:
a³ - b³ = (a - b)(a² + ab + b²)
Higher Power Identities
General Power of a Binomial:
(a + b)ⁿ = Σ [nCk * a^(n-k) * b^k], where 0 ≤ k ≤ n and nCk is the binomial coefficient.
Fourth Power Expansion:
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
Special Cases
Perfect Square Trinomials:
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
Three-Term Product:
(a + b + c)(a + b - c) = (a + b)² - c²
Solved Algebraic Identities with Examples
Question 1) Find the product of (x-1) (x-1)
Solution) We need to find the product (x-1) (x-1),
(x-1) (x-1) can also be written as (x-1)2.
We know the formula for (x-1)2, expand it
(a-b)2 = a2- 2ab+b2 where a= x, b=1
(x-1)2 = x2- 2x+1
Therefore, the product of (x-1) (x-1) is x2- 2x+1
Question 2) Find the product of (x+1) (x+1) as well as the value of it using x = 2.
Solution) We need to find the product (x+1) (x+1),
(x+1) (x+1) can also be written as (x+1)2.
We know the formula for (x+1)2, expand it
(a+b)2 = a2+ 2ab+b2 where a= x, b=1
(x+1)2 = x2+ 2x+1
Putting the value of x = 2 in equation 1,
(2)2+ 2(2) +1 = 9
Therefore, the product of (x+1) (x+1) is x2+ 2x+1 and the value of the expression is 9.
Question 3) Separate the constants and the variables from the given question.
-4, 4+x, 3x+4y, -5, 4.5y, 3y2+z
Solution) Variables are the ones which include any letter such as x, y, z etc along with the numbers.
In the given question,
Constants = -4, -5
Variables = 3x+4y, 4+x, 4.5y, 3y2+z
Question 4) Find the value of \[\frac{{{x^2} - 1}}{5}\],at x = -1.
Solution) At x = -1, \[x = - 1,\frac{{{x^2} - 1}}{5}\]
= \[\frac{{{(-1)^2} - 1}}{5}\]
= 0
Question 5) Find the value of x2+y2 – 10 at x=0 and y=0?
Solution) At x= 0 and y = 0,
x2+y2 – 10 = (0)2+(0)2 – 10
= -10
Question 6) Solve the following (x+2)2 using the concept of identities.
Solution) According to the identities and algebraic expression class 8,
We know the formula,
(a+b)2 = a2+2ab+b2
Where, a= x, b= 2
Let’s expand the given (x+2)2,
Therefore, (x+2)2 = x2+4x+4 is the solution.
FAQs on Algebraic Identities
1. How Many Identities are there in Algebraic Expressions?
The algebraic identities consists of three important identities. They are listed below-
(a+b)2 = a2+2ab+b2 |
(a-b)2 = a2- 2ab+b2 |
a2-b2= (a+b) (a-b) |
2. Give the Difference Between Algebraic Identity and Expression?
An algebraic identity is equality which is true for all the values whereas an expression which consists of variables and constants is known as an algebraic expression. The value of the expression changes every time the values are changed.
3. What is an Algebra Formula?
In mathematics, algebra is a combination of both numbers as well as letters. In the algebra formula the numbers remain fixed as their value is known and the letters or alphabets are used to represent unknown quantities which need to be found out.
4. What is the difference between algebraic expression and Identity?
An algebraic expression is an expression that consists of variables and constants. A variable can have any value in an expression. Thus, the value of the expression can change if the value of the variable is changed. But an algebraic identity is an equation that holds for all values of a variable. Materials on the same can also be found on the website of Vedantu which provides reliable material for all the students.
5. What is an algebraic formula?
When we started studying math as students, it was all about numbers. Natural numbers, integers, integers. Then we started to learn about mathematical functions like addition, subtraction, BODMAS and so on. As we go to higher classes, 8th grade, there are alphabets and letters in math. This is how our introduction to algebra began. In mathematics, algebra is a combination of numbers and letters. In algebraic formulas, the numbers remain constant/value is known and the letter or alphabet indicates the unknown number.
6. How many identities are in an algebraic expression?
An algebraic identity is an equation where the value of the left side of the equation is the same as the value of the right side of the equation. Unlike algebraic expressions, algebraic identifiers satisfy all variable values. The algebra identities especially help to solve many math problems. An algebraic identity is an algebraic equation that is always true for all values of the variables in it. Algebraic identifiers can be used to factor polynomials. They contain variables and constants on both sides of the equation. In algebraic identity, the left side of the equation is the same as the right side of the equation.
The category 8 algebraic identifier consists of three important identifiers. We list them below -
(a+b)2= a2+2ab+b2
(a-b)2 = a2-2ab+b2
a2-b2= (a+b) (a-b)
7. How to check algebraic identifiers?
Algebraic identification can be easily verified in two ways. One method is substitution math in which we replace the value to find the variable in the algebraic identifier. Algebraic identities have multiple expressions on both sides of the signed equation. Here we can substitute the values on both sides of the equation and try to get the same answer on both sides. Another method is the algebraic solution to verify the algebraic identity by optimizing and simplifying the left side of the equation to get the right side of the equation. This method requires geometric knowledge and certain materials to prove one's identity.
8. How do you memorize algebraic identities?
In algebra, numbers are replaced with letters of the alphabet to get a solution. These letters, for example (x, a, b, etc.) are used to represent unknown quantities in the equation. We then solve the equation or algebraic formula to get a definite answer. Algebraic identifiers are easy to remember by displaying the identifier as a box or rectangle. They can easily be remembered by factored form rather than the simplified form. The algebraic identities are the first stepping stone into the vast world of algebra. In order to excel in algebra, it is essential that the students know in detail about these identities otherwise it can be difficult to solve equations or math problems.
9. What Are Algebraic Identities?
Algebraic identities are mathematical equations that are always true, regardless of the values of the variables involved. These identities are used to simplify algebraic expressions and solve equations efficiently.
10. How Many Algebraic Identities Are There?
In basic algebra, there are three fundamental identities:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
11. What Are the Basic Algebraic Identities?
The basic algebraic identities include:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
These identities are widely used in algebra to simplify and factorize expressions.
12. What Are Some Examples of Algebraic Identities?
Example 1: For a = 2 and b = 3, verify (a + b)² = a² + 2ab + b².
Left-hand side: (2 + 3)² = 25
Right-hand side: 2² + 2(2)(3) + 3² = 25
Both sides are equal, confirming the identity.
13. How Do Algebraic Identities Differ From Algebraic Expressions?
An algebraic identity is an equation true for all values of the variables, while an algebraic expression is a combination of variables and constants without equality. The value of an expression changes based on the variables.
14. What Are Algebraic Identities Formulas?
Common formulas include:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
