factorbyinspection

Factor by inspection is a lightweight technique used to factor algebraic expressions or integers by examining their structure and applying common factoring patterns rather than performing systematic algorithms. It relies on recognizing patterns such as a shared factor, special products, or straightforward binomial or trinomial factorisations.

For polynomials, factor by inspection means looking for factors directly from the form and coefficients. Examples include x^2 + 5x + 6, which factors to (x+2)(x+3) because 2 and 3 multiply to 6 and add to 5. The difference of squares, x^2 − 9, factors as (x−3)(x+3). A perfect-square trinomial like x^2 + 6x + 9 factors as (x+3)^2. If a common factor is present, it is factored out first, as in 6x^2 + 9x = 3x(2x+3).

For integers, factor by inspection involves identifying a pair of factors whose product equals the number and, if needed, whose combination matches a given condition (such as a specific sum or parity). Examples: 45 = 5×9 or 3×15; 100 = 10×10 or 4×25. Divisibility tests (by 2, 3, 5, etc.) help guide which small primes to try first.

Limitations and scope: factor by inspection is not guaranteed to succeed and can miss factors that are not immediately obvious. It is most reliable for simple polynomials, low-degree cases, or problems designed with obvious factorable structure. For more complex expressions, systematic methods such as the AC method, factoring by grouping, the quadratic formula, the rational root theorem, or computational tools are used.