Special factoring patterns and factoring ax^2 + bx + c
Some polynomials match special patterns that make factoring fast. A difference of two squares factors as a^2 - b^2 = (a + b)(a - b), so x^2 - 49 = (x + 7)(x - 7) and 4x^2 - 9 = (2x + 3)(2x - 3). A perfect square trinomial factors as a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2, so x^2 + 6x + 9 = (x + 3)^2 and x^2 - 10x + 25 = (x - 5)^2. A sum of two squares, like x^2 + 9, cannot be factored using real numbers. Always factor out a greatest common factor first: 2x^2 - 18 = 2(x^2 - 9) = 2(x + 3)(x - 3). When the leading coefficient is not 1, look for factors that produce both the first and last terms and the correct middle term, as in 2x^2 + 7x + 3 = (2x + 1)(x + 3); you can check any factoring by multiplying back out.
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