How is a trinomial typically factored using the FOIL method?

How is a trinomial typically factored using the FOIL method?

• A. Factor out the greatest common factor only.
• B. Square the expression and take square roots.
• C. Set up a product of two binomials and find the factors that multiply to the last term and add to the middle term.
• D. Replace x with a constant and simplify.

Answer: (C)

FOIL works by multiplying two binomials and producing a middle term from the sum of their outer and inner parts. When you factor a trinomial by reversing that idea, you look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term. Those two numbers become the constants in the two binomials, giving you the factorization as a product of two binomials. So you end up with something like (x + r)(x + s) where r·s equals the last term and r + s equals the middle coefficient. For example, x^2 + 7x + 12 factors to (x + 3)(x + 4) because 3·4 = 12 and 3 + 4 = 7. This is the strategy you use to factor trinomials with leading coefficient 1 via FOIL. If the leading coefficient isn’t 1, you adjust using the same idea by considering ac, the two numbers that multiply to ac and add to b, then factor by grouping.

FOIL works by multiplying two binomials and producing a middle term from the sum of their outer and inner parts. When you factor a trinomial by reversing that idea, you look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term. Those two numbers become the constants in the two binomials, giving you the factorization as a product of two binomials. So you end up with something like (x + r)(x + s) where r·s equals the last term and r + s equals the middle coefficient. For example, x^2 + 7x + 12 factors to (x + 3)(x + 4) because 3·4 = 12 and 3 + 4 = 7. This is the strategy you use to factor trinomials with leading coefficient 1 via FOIL. If the leading coefficient isn’t 1, you adjust using the same idea by considering ac, the two numbers that multiply to ac and add to b, then factor by grouping.