Basic Expansion Formulas
In mathematics, expansion formulas are used to rewrite expressions in a simpler or more convenient form. These formulas are essential for simplifying complex expressions, solving equations, and understanding various mathematical concepts. Here are some basic expansion formulas:
1. Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form $(x + y)^n$, where n is a non-negative integer.
Formula: $(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$
where $\binom{n}{k}$ is the binomial coefficient, which is calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Example: Expanding $(x + y)^3$ using the binomial theorem:
$(x + y)^3 = \binom{3}{0} x^3 y^0 + \binom{3}{1} x^2 y^1 + \binom{3}{2} x^1 y^2 + \binom{3}{3} x^0 y^3$ $= x^3 + 3x^2y + 3xy^2 + y^3$
2. Sum and Difference of Cubes
These formulas allow us to expand expressions involving cubes.
Formula:
- Sum of Cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
- Difference of Cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
Example:
- Expanding $x^3 + 8$ using the sum of cubes formula: $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$
- Expanding $27y^3 - 1$ using the difference of cubes formula: $27y^3 - 1 = (3y - 1)(9y^2 + 3y + 1)$
3. Perfect Square Trinomials
These formulas simplify expressions involving squares of binomials.
Formula:
- Square of a Sum: $(a + b)^2 = a^2 + 2ab + b^2$
- Square of a Difference: $(a - b)^2 = a^2 - 2ab + b^2$
Example:
- Expanding $(2x + 3y)^2$ using the square of a sum formula: $(2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2 = 4x^2 + 12xy + 9y^2$
- Expanding $(5x - 2y)^2$ using the square of a difference formula: $(5x - 2y)^2 = (5x)^2 - 2(5x)(2y) + (2y)^2 = 25x^2 - 20xy + 4y^2$
4. Difference of Squares
This formula is used to factor expressions involving the difference of two squares.
Formula: $a^2 - b^2 = (a + b)(a - b)$
Example:
- Expanding $9x^2 - 4y^2$ using the difference of squares formula: $9x^2 - 4y^2 = (3x + 2y)(3x - 2y)$
These basic expansion formulas are valuable tools for simplifying and manipulating algebraic expressions. Mastering these formulas will enhance your understanding of algebra and other related mathematical concepts.
