Trinomials with Lead Coefficients

Students learn that a trinomial in the form ax^2 + bx + c (where c is positive), such as 3x^2 – 22x + 7, can be factored as the product of two binomials, in this case (3x - 1)(x – 7). The first term in each binomial comes from the factors of 3x^2, 3x and x. The second term in each binomial comes from the factors of the constant term, +7, such that the product of the outer terms plus the product of the inner terms equals the middle term, –22x. Note that if -1 and –7 are used as the second terms in the binomials (3x - 1)(x – 7), then the product of the outer terms, -21x, plus the product of the inner terms, -1x, is –22x, which equals the original middle term.