Algebra - Factorization of Algebraic Expressions

Factorization as the Reverse of Expansion: To factorize an algebraic expression means to write it as a product of its factors. If you imagine a large rectangle where the total area is the algebraic expression, factorizing is the process of finding the specific lengths of its sides. For example, $ab + ac$ represents the area, and $a(b + c)$ represents the length and width.

Common Factor Extraction: This is the primary step where you identify the Highest Common Factor (HCF) of all terms in the expression and place it outside a bracket. Visually, this is like identifying a common width for multiple rectangular areas and 'pulling' it out to see the remaining lengths.

Difference of Two Squares (DOTS): This pattern occurs when a perfect square is subtracted from another, such as $a^2 - b^2$. Visually, if you take a large square of side $a$ and remove a smaller square of side $b$ from the corner, the remaining shape can be sliced and rearranged into a single rectangle with dimensions $(a+b)$ and $(a-b)$.

Factorizing by Grouping: This technique is often used for expressions with four terms. You group the terms into two pairs and factor out the HCF from each pair. If the resulting binomials in the brackets are identical, they become a single common factor. Visually, this is like organizing four separate area blocks into two distinct rows that eventually align to form a single large grid.

Factoring Monic Quadratic Trinomials: For expressions like $x^2 + bx + c$, you look for two integers that multiply to give $c$ (the product) and add to give $b$ (the sum). Visually, this is the challenge of taking one $x^2$ tile, $b$ number of $x$ tiles, and $c$ unit squares and arranging them into a perfectly solid rectangle without any gaps.

Perfect Square Trinomials: These are special trinomials that factor into a squared binomial, such as $(a+b)^2$ or $(a-b)^2$. You can identify them because the first and last terms are perfect squares, and the middle term is exactly twice the product of the square roots of the outer terms. Visually, these pieces always form a perfect square rather than a non-square rectangle.