Algebra - Expanding and factorizing algebraic expressions, including quadratic expressions

The Distributive Law: This concept involves multiplying a single term outside a bracket by every term inside the bracket, represented as $a(b + c) = ab + ac$. Visually, this can be understood as finding the area of a large rectangle with width $a$ and length $(b+c)$, which is equivalent to the sum of the areas of two smaller rectangles, $ab$ and $ac$.

Expanding Binomials (FOIL): To expand an expression like $(a + b)(c + d)$, we multiply the First terms, Outer terms, Inner terms, and Last terms. This is often visualized using a 2x2 area model or grid, where each of the four cells contains the product of one term from each binomial, which are then summed together to get $ac + ad + bc + bd$.

Common Factorizing: This is the inverse process of expansion. It involves identifying the Highest Common Factor (HCF) of all terms in an expression and placing it outside a set of brackets. For example, in $4x^2 + 8x$, the HCF is $4x$, allowing the expression to be rewritten as $4x(x + 2)$. This is like dividing the total area of a rectangle by a known side length to find the remaining side.

Difference of Two Squares: A specific pattern where two perfect squares are subtracted, written as $a^2 - b^2$. Visually, if you take a square of area $a^2$ and remove a smaller square of area $b^2$ from its corner, the remaining shape can be sliced and rearranged into a rectangle with side lengths $(a - b)$ and $(a + b)$, giving the identity $(a - b)(a + b)$.

Perfect Square Trinomials: Expanding $(a + b)^2$ or $(a - b)^2$ results in $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$ respectively. Visually, $(a+b)^2$ represents a square with side length $(a+b)$. Inside this square, you will see one square of area $a^2$, one square of area $b^2$, and two rectangles each with area $ab$.

Factorizing Trinomials ($x^2 + bx + c$): To factorize a quadratic where the $x^2$ coefficient is 1, we look for two numbers that multiply to give the constant $c$ and add to give the coefficient $b$. Visually, this is the process of arranging $x^2$ tiles, $x$ tiles, and unit tiles into a single large rectangle where the side lengths represent the factors $(x+p)$ and $(x+q)$.

Factorizing by Grouping: This technique is applied to expressions with four terms, such as $ax + ay + bx + by$. We group the terms into two pairs that share common factors: $a(x + y) + b(x + y)$. Since $(x + y)$ is now a common factor for both groups, it can be factored out to produce $(a + b)(x + y)$.