Number and Algebra - Algebraic manipulation (expansion, factorization)

Expansion using the Distributive Law: This process involves multiplying a term outside a bracket by every term inside, such as $a(b + c) = ab + ac$. For binomials like $(a + b)(c + d)$, we use the FOIL method (First, Outer, Inner, Last). Visually, expansion can be viewed as finding the total area of a large rectangle that has been partitioned into smaller sections.

Factorization by Highest Common Factor (HCF): This is the reverse of expansion. It involves identifying the largest factor that divides all terms in an expression and placing it outside a set of brackets. For example, in $4x^2 + 8x$, the HCF is $4x$, resulting in $4x(x + 2)$. Visually, this is like identifying a common dimension shared by several rectangular blocks.

Difference of Two Squares (DOTS): An expression in the form $a^2 - b^2$ can always be factorized into $(a - b)(a + b)$. Visually, this represents taking a square of area $a^2$, removing a smaller square of area $b^2$ from its corner, and rearranging the remaining L-shaped region into a single rectangle with sides $(a - b)$ and $(a + b)$.

Perfect Square Trinomials: These are specific patterns where $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$. Visually, $(a+b)^2$ represents a square with side length $a+b$, which is composed of a square of area $a^2$, a square of area $b^2$, and two identical rectangles each with area $ab$.

Factorizing Quadratic Trinomials ($x^2 + bx + c$): To factorize a trinomial where the leading coefficient is $1$, find two numbers that multiply to give the constant term $c$ and add to give the coefficient of the middle term $b$. These numbers, $p$ and $q$, allow the expression to be written as $(x + p)(x + q)$. On a coordinate plane, these values relate to the $x$-intercepts of the parabola.

Factorization by Grouping: This technique is used for expressions with four terms, such as $ax + ay + bx + by$. Terms are grouped into pairs, a common factor is extracted from each pair, and then the common binomial factor is extracted. For example, $a(x + y) + b(x + y)$ becomes $(a + b)(x + y)$. This can be visualized as splitting a complex composite shape into two simpler rectangular groups.

Simplifying Algebraic Fractions: Algebraic fractions are simplified by first factorizing the numerator and the denominator completely. Once factorized, common binomial or monomial factors that appear in both the top and bottom can be cancelled out (divided to equal $1$). This is visually similar to reducing a numerical fraction by dividing out common prime factors.