Productos Notables en Álgebra

Study Notes

Notable Products in Algebra

Difference of Squares: (a - b)(a + b) = a² - b²
- Represents the product of a binomial difference and a binomial sum as a difference of squares.
- Fundamental for factoring expressions fitting this pattern.
Square of a Sum: (a + b)² = a² + 2ab + b²
- Describes how the square of a binomial sum expands.
- Requires adding twice the product of the terms to the sum of their squares.
Square of a Difference: (a - b)² = a² - 2ab + b²
- Similar to the square of a sum, but subtracts twice the product of the terms from the sum of their squares.
Product of a Sum and a Difference: (a + b)(a - b) = a² - b²
- A particular instance of the difference of squares formula.
- Useful in factoring expressions where identical terms appear as a sum and a difference.
Cube of a Sum: (a + b)³ = a³ + 3a²b + 3ab² + b³
- A more intricate expansion for the cube of a binomial sum.
- The terms incorporate binomial coefficients aligning with the third row of Pascal's triangle.
Cube of a Difference: (a - b)³ = a³ - 3a²b + 3ab² - b³
- The cube of a binomial difference, corresponding to the cube of a sum.
- The signs of the terms alternate.
Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
- Essential factorization for the sum of two perfect cubes.
- The factorization contains a quadratic term.
Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
- Crucial for factoring the difference of two perfect cubes.
- Another factorization formula with a quadratic term.
General Binomial Expansion: (a + b)ⁿ = ∑ (n choose k) a^(n-k) b^k (for any non-negative integer n)
- The binomial theorem, representing any positive integer power of (a + b) through binomial coefficients ("n choose k").
- Involves all possible values of the exponent of 'b'.
Important Note: These formulas are essential tools for algebraic manipulations, enabling efficient factoring and expanding of expressions. Precise application of these formulas directly impacts the success of problem-solving.