Special Product Formulas
*Notice: Perfect squares on both ends of the trinomial!
Example: Factor: .
Since the product of two binomials is often a trinomial, we expect that many trinomials will factor as two binomials.
But not all…
Test for Factorability
A trinomial of the form with integer coefficients and a ≠0, will factor into two binomials with integer coefficients if the value of is a perfect square. If , the factors will be the same.
Example 4: Test for factorability:
Example 5: Test for factorability:
The AC Method
Example 1: Factor:
Always first test for factorability:
For each factorable trinomial of the form , multiply the factors a and c. Then list all possible factorizations of ac.
Choose the factorization where the sum of the factors is b. Then rewrite the trinomial using the chosen factors instead of b.
Factor by grouping.
Example 2: Factor:
Begin by writing the polynomial in the form . Then test for factorability:
Example 3: Factor: .
When possible we will always want to factor out a common factor before using the ac method and before testing for factorability.
Example 6: Factor: .
Example 7: Factor: .
Using Substitution to Factor Trinomials
Example 9: Factor:
Factoring by Grouping
Example 10: Factor: