What is an Expansion?
In algebra, an expansion is the process of multiplying out expressions that are enclosed within brackets to form a single, continuous expression without brackets. This is achieved by applying the distributive law of multiplication. Instead of multiplying long-form expressions manually every time, we use shorthand formulas called algebraic identities. These identities act like mathematical shortcuts.
Think of expansions like unpacking a tightly wrapped parcel. For example, if you have a square plot of land whose sides are increased, calculating the new area visually is exactly what the expansion of a squared binomial represents.
Let us look at the fundamental identities for squares and cubes of binomials (two-term expressions):
- Square of a Sum: When you square the sum of two numbers, it expands into the square of the first number, plus twice the product of both numbers, plus the square of the second number.
- Square of a Difference: This is similar to the sum, except the middle term becomes negative because a positive multiplied by a negative results in a negative value.
- Cube of a Sum: When an expression is raised to the power of three, it describes a volumetric expansion containing cubic and rectangular blocks.
- Cube of a Difference: Symmetrical to the cubic sum, with alternating mathematical signs.
Here is a summary table of these fundamental expansions:
| Binomial Form | Expanded Polynomial Structure | Alternative Factored Variant |
|---|---|---|
| **(a - b)²** | a² - 2ab + b² | a² + b² - 2ab |
| **(a + b)³** | a³ + 3a²b + 3ab² + b³ | a³ + b³ + 3ab(a + b) |
| **(a - b)³** | a³ - 3a²b + 3ab² - b³ | a³ - b³ - 3ab(a - b) |