Algebra Class: Binomial Theorems

Study Notes

Algebraic Identities

Square of a Sum: ((a + b)^2 = a^2 + 2ab + b^2) is equivalent to ((-a - b)^2).
Square of a Difference: ((a - b)^2 = a^2 - 2ab + b^2).
Difference of Squares: ((a - b)(a + b) = a^2 - b^2).

Expanding Squares with Three Variables

Square of a Sum with Three Variables: ((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca).
Square of a Two-Variable Sum with a Negative Variable: ((a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2bc - 2ca).
Square with One Negative Variable: ((a - b + c)^2 = a^2 + b^2 + c^2 - 2ab - 2bc + 2ca).
Negative Variable in the First Position: ((-a + b + c)^2 = a^2 + b^2 + c^2 - 2ab + 2bc - 2ca).
All Negative Variables: ((a - b - c)^2 = a^2 + b^2 + c^2 - 2ab + 2bc - 2ca).

Cube Expansions

Cube of a Sum: ((a + b)^3 = a^3 + b^3 + 3ab(a + b)).
Cube of a Difference: ((a - b)^3 = a^3 - b^3 - 3ab(a - b)).

Factoring Sums and Differences of Cubes

Sum of Cubes: (a^3 + b^3 = (a + b)^3 - 3ab(a + b)) can also be expressed as ((a + b)(a^2 - ab + b^2)).
Difference of Cubes: (a^3 - b^3 = (a - b)^3 + 3ab(a - b)) or ((a - b)(a^2 + ab + b^2)).

Special Identity involving Three Variables

Cubic Form: (a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)).
- If (a + b + c = 0), then it leads to (a^3 + b^3 + c^3 = 3abc).

Description

Test your knowledge on binomial expansions and identities in this algebra class quiz. Covering essential formulas and properties, this quiz will help you reinforce your understanding of algebraic expressions involving squares and cubes. Prepare to solve equations and recognize patterns involving variables a, b, and c.