Algebra - Expansions

Expansion is the algebraic process of removing brackets by multiplying terms. Visually, the expansion $(a+b)(c+d)$ can be represented as the area of a large rectangle with sides $(a+b)$ and $(c+d)$, which is equal to the sum of the areas of four smaller internal rectangles: $ac, ad, bc,$ and $bd$.

Squaring a binomial $(a+b)^2$ results in a trinomial. Geometrically, this represents a square of side $a+b$ being divided into four sections: a square of area $a^2$, a square of area $b^2$, and two identical rectangles each of area $ab$. The result is $a^2 + 2ab + b^2$.

The product of a sum and a difference, $(a+b)(a-b)$, leads to the 'Difference of Two Squares' identity. Visually, if you take a square with side $a$ (area $a^2$) and remove a smaller square of side $b$ (area $b^2$) from the corner, the remaining shape can be sliced and rearranged into a rectangle with dimensions $(a+b)$ and $(a-b)$.

Expansion of a trinomial $(a+b+c)^2$ involves the sum of the squares of each term plus twice the product of every possible pair of terms. This expands into 6 terms: $a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.

Cubing a binomial $(a+b)^3$ represents the volume of a cube with side length $a+b$. This volume is composed of a smaller cube $a^3$, another cube $b^3$, three rectangular prisms with dimensions $a \times a \times b$, and three rectangular prisms with dimensions $a \times b \times b$.

Expansion of reciprocals involves expressions like $(x + \frac{1}{x})^2$. Because the product of a number and its reciprocal is 1 ($x \cdot \frac{1}{x} = 1$), the middle term of the expansion becomes a constant integer, which is a unique property used to solve higher-order power problems.

Conditional identities are specific results that hold true under given constraints. A common one is: if $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$. This simplifies calculations for sum of cubes significantly when the sum of the bases is zero.