With binomials, students expand the terms with the common Foil method. The process for this method involves multiplying the first terms, then the outside terms, the inside terms, and finally the last terms. However, the Foil method is useless for expanding trinomials because although you can multiply the first terms, the inside and last terms overlap, and if you multiply per the Foil method, you remove one of the factors necessary to come up with the correct solution. In addition, the products of the terms are quite lengthy and the chances of mathematical errors are great.
Examine the trinomial (x + 3)(x + 4)(x + 5).
Multiply the first two binomials using the distributive property. (x) x (x) = x^2, (x) x (4) = 4x, (3) x (x) = 3x and (3) x (4) = 12. You should have a polynomial that reads x^2 + 4x + 3x + 12.
Combine like terms: x^2 + (4x + 3x) + 12 = x^2 + 7x + 12.
Multiply the new trinomial by the last binomial from the original problem with the distributive property: (x + 5)(x^2 + 7x + 12). (x) x (x^2) = x^3, (x) x (7x) = 7x^2, (x) x (12) = 12x, (5) x (x^2) = 5x^2, (5) x (7x) = 35x and (5) x (12) = 60. You should have a polynomial that reads x^3 + 7x^2 + 12x + 5x^2 + 35x + 60.
Combine like terms: x^3 + (7x^2 + 5x^2) + (12x + 35x) + 60 = x^3 + 12x^2 + 47x + 60.