2. 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

3. Example continued Factor: x2 + 7x + 12
Since 3  4 = 12 and = 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

5. -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)

6. -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)

7. 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.

10. 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.

12. 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.