Standard Identities
What is Standard Identities?
An equation that is true for all possible values of the variables involved.
Key formula / rule: Square of a binomial sum
Key points
- Understand the concept of algebraic identities.
- Memorize and correctly apply standard algebraic identities.
- Use identities to simplify algebraic expressions.
- Solve problems involving sums, differences, and products of algebraic terms.
Common exam trap
Confusing (a+b)2 with a2 + b2.
Definitions
- Term
Algebraic Identity
- Meaning
An equation that is true for all possible values of the variables involved.
- Term
Binomial
- Meaning
An algebraic expression consisting of two terms.
- Term
Trinomial
- Meaning
An algebraic expression consisting of three terms.
Learning objectives
Understand the concept of algebraic identities.
Memorize and correctly apply standard algebraic identities.
Use identities to simplify algebraic expressions.
Solve problems involving sums, differences, and products of algebraic terms.
Apply identities in number theory and geometry problems.
Formulae
- Name
Square of a binomial sum
- Note
Used to expand or factor expressions involving squares of sums.
- Expression
(a+b)2 = a2 + 2ab + b2
- Name
Square of a binomial difference
- Note
Used to expand or factor expressions involving squares of differences.
- Expression
(a-b)2 = a2 - 2ab + b2
- Name
Difference of squares
- Note
Crucial for factoring quadratic expressions and simplifying fractions.
- Expression
a2 - b2 = (a+b)(a-b)
- Name
Sum of cubes
- Note
Used for factoring cubic sums.
- Expression
a3 + b3 = (a+b)(a2 - ab + b2)
- Name
Difference of cubes
- Note
Used for factoring cubic differences.
- Expression
a3 - b3 = (a-b)(a2 + ab + b2)
- Name
Alternative sum of cubes
- Note
Useful when sum (a+b) and product ab are known.
- Expression
a3 + b3 = (a+b)3 - 3ab(a+b)
- Name
Alternative difference of cubes
- Note
Useful when difference (a-b) and product ab are known.
- Expression
a3 - b3 = (a-b)3 + 3ab(a-b)
- Name
Square of a trinomial
- Note
Used for expanding squares of expressions with three terms.
- Expression
(a+b+c)2 = a2 + b2 + c2 + 2(ab+bc+ca)
Prerequisites
Basic arithmetic operations.
Understanding of variables and constants.
Basic laws of exponents.
Factorization concepts.
Common mistakes
Confusing (a+b)2 with a2 + b2.
Incorrectly applying signs in identities like (a-b)2.
Memorizing formulas incorrectly, especially for cubic identities.
Errors in substituting values into the identities.
Keywords
Algebraic Identities
Standard Formulas
Simplification
Factorization
Quantitative Aptitude
SSC CGL
Polynomials
Cubic Identities
Quadratic Expressions
Practice preview
What is the value of (99)^2 - (98)^2?…
easy
If x + 1/x = 4, what is the value of x^2 + 1/x^2?…
easy
If a + b = 7 and ab = 12, what is the value of a^2 + b^2?…
medium