Solution Think of FOIL in reverse. (x + )(x + ) Example Solution Think of FOIL in reverse. (x + )(x + ) We need 2 constant terms that have a product of 12 and a sum of 7. We list some pairs of numbers that multiply to 12. Product Sum 7 3 4 2 6 Guess 12

Example continued Factor: x2 + 7x + 12 Since 3  4 = 12 and 3 + 4 = 7, the factorization of x2 + 7x + 12 is (x + 3)(x + 4). To check we simply multiply the two binomials. Check: (x + 3)(x + 4) = x2 + 4x + 3x + 12 = x2 + 7x + 12

-8 -3 -5 15 Example Factor: y2  8y + 15 Solution Since the constant term is positive and the coefficient of the middle term is negative, we look for the factorization of 15 in which both factors are negative. Their sum must be 8. Product Sum -8 -3 -5 Guess 15 y2  8y + 15 = (y  3)(y  5)

-5 2 -12 3 -8 4 -6 -24 Example Factor: x2  5x  24 Solution The constant term must be expressed as the product of a negative number and a positive number. Since the sum of the two numbers must be negative, the negative number must have the greater absolute value. Product Sum -5 2 -12 3 -8 4 -6 Guess -24 x2  5x  24 = (x + 3)(x  8)

4 -4 8 4 -8 t2 + 4t  32 = (t + 8)(t  4) -32 Example Solution We need one positive and one negative factor. The sum must be 4, so the positive factor must have the larger absolute value. Product Sum 4 -4 8 4 -8 Guess t2 + 4t  32 = (t + 8)(t  4) -32 Conclusion, the process of factoring trinomials is guess and check.

Often factoring requires two or more steps Often factoring requires two or more steps. Remember, when told to factor, we should factor completely. This means the final factorization should contain only prime polynomials.

Examples Factor Completely Often factoring requires two or more steps. Remember, when told to factor, we should factor completely. This means the final factorization should contain only prime polynomials.