Algebra
What is Algebra?
An equation that remains true for all possible values of the variables involved.
Key formula / rule: Square of a sum
Key points
- Understand the definition of an algebraic identity.
- Memorize and correctly apply common algebraic identities.
- Use identities to simplify algebraic expressions.
- Use identities for factorization.
Common exam trap
Confusing identities with conditional equations.
Definitions
- Term
Algebraic Identity
- Meaning
An equation that remains true for all possible values of the variables involved.
- Term
Binomial
- Meaning
An algebraic expression consisting of two terms.
- Term
Polynomial
- Meaning
An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- Term
Factorization
- Meaning
The process of expressing a polynomial as a product of its factors.
Learning objectives
Understand the definition of an algebraic identity.
Memorize and correctly apply common algebraic identities.
Use identities to simplify algebraic expressions.
Use identities for factorization.
Solve problems involving algebraic identities efficiently.
Formulae
- Name
Square of a sum
- Note
Used for expanding binomials and simplifying expressions.
- Expression
(a+b)² = a² + 2ab + b²
- Name
Square of a difference
- Note
Used for expanding binomials and simplifying expressions.
- Expression
(a-b)² = a² - 2ab + b²
- Name
Difference of squares
- Note
Crucial for factorization and simplification.
- Expression
a² - b² = (a-b)(a+b)
- Name
Sum of cubes
- Note
Used for factorization.
- Expression
a³ + b³ = (a+b)(a² - ab + b²)
- Name
Difference of cubes
- Note
Used for factorization.
- Expression
a³ - b³ = (a-b)(a² + ab + b²)
- Name
Cube of a sum
- Note
Used for expansion.
- Expression
(a+b)³ = a³ + 3a²b + 3ab² + b³
- Name
Cube of a difference
- Note
Used for expansion.
- Expression
(a-b)³ = a³ - 3a²b + 3ab² - b³
- Name
Sum of squares of three terms
- Note
Useful for simplifying expressions with three variables.
- Expression
(a+b+c)² = a² + b² + c² + 2(ab + bc + ca)
- Name
Factorization of sum of cubes (alternative)
- Note
Alternative form for sum of cubes factorization.
- Expression
a³ + b³ = (a+b)³ - 3ab(a+b)
- Name
Factorization of difference of cubes (alternative)
- Note
Alternative form for difference of cubes factorization.
- Expression
a³ - b³ = (a-b)³ + 3ab(a-b)
- Name
Identity involving sum of cubes and abc
- Note
Important for problems involving sums of cubes.
- Expression
a³ + b³ + c³ - 3abc = (a+b+c)(a² + b² + c² - ab - bc - ca)
Prerequisites
Basic arithmetic operations.
Understanding of variables and constants.
Basic algebraic expressions and equations.
Common mistakes
Confusing identities with conditional equations.
Incorrectly applying signs in expansions (e.g., (a-b)² = a² - b²).
Forgetting the middle term (2ab) in square expansions.
Mistakes in factorization using identities.
Keywords
Algebraic Identities
Polynomials
Factorization
Expansion
Quadratic Equations
Cubic Equations
Simplification
Practice preview
If x + y = 5 and xy = 6, what is the value of x^3 + y^3?…
medium
Solve for x: 3x + 8 = 20.…
easy
Find the roots of the quadratic equation x^2 - 5x + 6 = 0.…
medium