Algebraic Identities: Examples and Chart

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Algebraic Identities Examples and Chart

Algebraic identities are very important in algebra. They are special equations that stay true no matter what numbers you use for the variables. Furthermore, unlike regular equations with specific answers, these identities always hold true. Additionally, they are like the building blocks of working with expressions and making tough equations simpler for us to solve. Read on to learn more in detail about Algebraic identities, Examples and a Chart of Algebraic identities. 

Also Read: Algebra Questions

Examples of Algebraic Identities

In addition, here are some of the most common Algebraic identities:

  • Difference of Squares: This rule says that if you square a binomial (which is a sum of two terms), it equals the square of the first term plus twice the product of the first and second terms, minus the square of the second term. 
    • It looks like this: (a + b)² = a² + 2ab + b²
  • Perfect Square Trinomial: This rule talks about expanding a trinomial where the first two terms are perfect squares.
    • It says that this type of trinomial can be written as the square of a binomial. 
    • The rule is: (a + b)² + 2c(a + b) + c² = (a + b + c)²
  • Sum-Product Pattern: This rule connects the sum of two numbers to their product.
    • Moreover, it says that if you add two numbers and multiply them by their difference, it equals the difference of their squares. 
    • It is written as: (a + b)(a – b) = a² – b²
  • Zero Product Property: This is a basic rule saying that if the product of two things equals zero, then at least one of them must be zero. 
    • It is written as: ab = 0 => a = 0 or b = 0 (or both)
  • Commutative Property: This property says that for addition and multiplication, changing the order doesn’t change the result. 
    • For Addition: a + b = b + a
    • For Multiplication: ab = ba

However, these are just a few examples of algebraic rules used in math.

Also Read: Understanding Set Theory Formulas & Questions

Chart of Algebraic Identities

Furthermore, here is a Chart of Algebraic Identities:

Chart of Algebraic Identities 
Identity Name FormulaDescriptionExample
Difference of Squares(a + b)² = a² + 2ab + b²Square of a binomialExpand (x + 2)²: (x + 2)² = x² + (2)(x)(2) + 2² = x² + 4x + 4
Perfect Square Trinomial(a + b)² + 2c(a + b) + c² = (a + b + c)²Expansion of a specific trinomialExpand (x + 1)² + 4(x + 1) + 4: (x + 1)² + 4(x + 1) + 4 = (x + 1 + 2)² = (x + 3)²
Sum-Product Pattern(a + b)(a – b) = a² – b²Relates sum and product of variablesFind the product of (x + y) and (x – y): (x + y)(x – y) = x² – (y²)
Zero Product Propertyab = 0 => a = 0 or b = 0If the product is zero, one or both factors must be zeroSolve the equation xy = 0: According to the zero product property, either x = 0 or y = 0 (or both).
Commutative Propertya + b = b + a (for addition)The order does not affect addition5 + 3 = 3 + 5 (both equal 8)
Commutative Propertyab = ba (for multiplication)Order does not affect the multiplication2 x 7 = 7 x 2 (both equal 14)
Difference of Cubes(a – b)³ = a³ – 3a²b + 3ab² – b³Cube of a binomial with a differenceSimplify (x – 2)³: (x – 2)³ = x³ – 3(x²)(2) + 3(x)(2²) – 2³
Binomial Theorem(a + b)^n = a^n + n(a^(n-1))b + … + b^n (where n is a non-negative integer)Expansion of a binomial raised to any powerExpand (x + y)⁴ using the binomial theorem.
Chart of Algebraic Identities

How many Algebraic Identities are there?

Algebraic identities are like tools in a toolbox for math. While we cannot exactly count how many there are, there are tons of them! They come from doing different math stuff, like adding, subtracting, multiplying, and dealing with different numbers in equations. However, there are basic ones that are very important. Interestingly, math people are always discovering or making new identities as they explore math more deeply.

Additionally, these identities are useful because they can do a lot of different jobs:

  • Making Math Easier: We can use them to simplify complicated math problems by spotting patterns and using these special rules.
  • Breaking Down Big Math Problems: Moreover, some identities help us break down big, complicated expressions into smaller, easier ones. This is very helpful when we are trying to solve equations.
  • Creating New Math Rules: Whilst playing around with the ones we already know about and putting them together in different ways, we can come up with brand-new rules that help us solve even more challenging problems, over the years. 

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I hope this helps! Did you like learning about Algebraic Identities? Keep reading our blogs to learn more about the basic concepts of Maths!

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